Topics
Surrounding
the
Combinatorial
Anabelian
Geometry
of
Hyperbolic
Curves
I:
Inertia
Groups
and
Profinite
Dehn
Twists
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Received
March
31,
2011.
Revised
December
28,
2011.
2010
Mathematics
Subject
Classification.
Primary
14H30;
Secondary
14H10.
Key
words
and
phrases.
anabelian
geometry,
combinatorial
anabelian
geom-
etry,
profinite
Dehn
twist,
semi-graph
of
anabelioids,
inertia
group,
hyperbolic
curve,
configurationa
space.
The
first
author
was
supported
by
Grant-in-Aid
for
Young
Scientists
(B),
No.
22740012,
Japan
Society
for
the
Promotion
of
Science.
2
Abstract.
Let
Σ
be
a
nonempty
set
of
prime
numbers.
In
the
present
pa-
per,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyper-
bolic
curves
and
their
associated
configuration
spaces
over
algebraically
closed
fields
of
characteristic
zero.
Our
first
main
result
asserts,
roughly
speaking,
that
if
an
F-admissible
automorphism
[i.e.,
an
automorphism
that
preserves
the
fiber
subgroups
that
arise
as
kernels
associated
to
the
various
natural
projections
of
the
configuration
space
under
consider-
ation
to
configuration
spaces
of
lower
dimension]
of
a
configuration
space
group
arises
from
an
F-admissible
automorphism
of
a
configura-
tion
space
group
[arising
from
a
configuration
space]
of
strictly
higher
dimension,
then
it
is
necessarily
FC-admissible,
i.e.,
preserves
the
cus-
pidal
inertia
subgroups
of
the
various
subquotients
corresponding
to
surface
groups.
After
discussing
various
abstract
profinite
combinato-
rial
technical
tools
involving
semi-graphs
of
anabelioids
of
PSC-type
that
are
motivated
by
the
well-known
classical
theory
of
topological
surfaces,
we
proceed
to
develop
a
theory
of
profinite
Dehn
twists,
i.e.,
an
abstract
profinite
combinatorial
analogue
of
classical
Dehn
twists
associated
to
cycles
on
topological
surfaces.
This
theory
of
profinite
Dehn
twists
leads
naturally
to
comparison
results
between
the
abstract
combinatorial
machinery
developed
in
the
present
paper
and
more
clas-
sical
scheme-theoretic
constructions.
In
particular,
we
obtain
a
purely
combinatorial
description
of
the
Galois
action
associated
to
a
[scheme-
theoretic!]
degenerating
family
of
hyperbolic
curves
over
a
complete
equicharacteristic
discrete
valuation
ring
of
characteristic
zero.
Finally,
we
apply
the
theory
of
profinite
Dehn
twists
to
prove
a
“geometric
ver-
sion
of
the
Grothendieck
Conjecture”
for
—
i.e.,
put
another
way,
we
compute
the
centralizer
of
the
geometric
monodromy
associated
to
—
the
tautological
curve
over
the
moduli
stack
of
pointed
smooth
curves.
Contents
Introduction
3
0.
Notations
and
Conventions
11
1.
F-admissibility
and
FC-admissibility
14
2.
Various
operations
on
semi-graphs
of
anabelioids
of
PSC-type
30
3.
Synchronization
of
cyclotomes
49
4.
Profinite
Dehn
multi-twists
74
5.
Comparison
with
scheme
theory
92
6.
Centralizers
of
geometric
monodromy
122
Combinatorial
anabelian
topics
I
3
§
Introduction
Let
Σ
⊆
Primes
be
a
nonempty
subset
of
the
set
of
prime
numbers
Primes.
In
the
present
paper,
we
continue
our
study
[cf.
[SemiAn],
[CmbGC],
[CmbCsp],
[MT],
[NodNon]]
of
the
anabelian
geometry
of
semi-graphs
of
anabelioids
of
[pro-Σ]
PSC-type,
i.e.,
semi-graphs
of
an-
abelioids
that
arise
from
a
pointed
stable
curve
over
an
algebraically
closed
field
of
characteristic
zero.
Roughly
speaking,
such
a
“semi-
graph
of
anabelioids”
may
be
thought
of
as
a
slightly
modified,
Galois
category-theoretic
formulation
of
the
“graph
of
profinite
groups”
asso-
ciated
to
such
a
pointed
stable
curve
that
takes
into
account
the
cusps
[i.e.,
marked
points]
of
the
pointed
stable
curve,
and
in
which
the
profi-
nite
groups
that
appear
are
regarded
as
being
defined
only
up
to
inner
automorphism.
At
a
more
conceptual
level,
the
notion
of
a
semi-graph
of
anabelioids
of
PSC-type
may
be
thought
of
as
a
sort
of
abstract
profi-
nite
combinatorial
analogue
of
the
notion
of
a
hyperbolic
topolog-
ical
surface
of
finite
type,
i.e.,
the
underlying
topological
surface
of
a
hyperbolic
Riemann
surface
of
finite
type.
One
central
object
of
study
in
this
context
is
the
notion
of
an
outer
representation
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)],
which
may
be
thought
of
as
an
abstract
profinite
combinatorial
analogue
of
the
scheme-theoretic
notion
of
a
de-
generating
family
of
hyperbolic
curves
over
a
complete
discrete
valuation
ring.
In
[NodNon],
we
studied
a
purely
combinatorial
generalization
of
this
notion,
namely,
the
notion
of
an
outer
representation
of
NN-type
[cf.
[NodNon],
Definition
2.4,
(iii)],
which
may
be
thought
of
as
an
abstract
profinite
combinatorial
analogue
of
the
topological
notion
of
a
family
of
hyperbolic
topological
surfaces
of
finite
type
over
a
circle.
Here,
we
recall
that
such
families
are
a
central
object
of
study
in
the
theory
of
hyperbolic
threefolds.
Another
central
object
of
study
in
the
combinatorial
anabelian
ge-
ometry
of
hyperbolic
curves
[cf.
[CmbCsp],
[MT],
[NodNon]]
is
the
no-
tion
of
a
configuration
space
group
[cf.
[MT],
Definition
2.3,
(i)],
i.e.,
the
pro-Σ
fundamental
group
of
the
configuration
space
associated
to
a
hyperbolic
curve
over
an
algebraically
closed
field
of
characteristic
zero,
where
Σ
is
either
equal
to
Primes
or
of
cardinality
one.
In
[MT],
it
was
shown
[cf.
[MT],
Corollary
6.3]
that,
if
one
excludes
the
case
of
hyperbolic
curves
of
type
(g,
r)
∈
{(0,
3),
(1,
1)},
then,
up
to
a
permu-
tation
of
the
factors
of
the
configuration
space
under
consideration,
any
automorphism
of
a
configuration
space
group
is
necessarily
F-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)],
i.e.,
preserves
the
fiber
subgroups
that
arise
as
kernels
associated
to
the
various
natural
projections
of
the
4
Yuichiro
Hoshi
and
Shinichi
Mochizuki
configuration
space
under
consideration
to
configuration
spaces
of
lower
dimension.
In
§1,
we
prove
our
first
main
result
[cf.
Corollary
1.9],
by
means
of
techniques
that
extend
the
techniques
of
[MT],
§4,
i.e.,
techniques
that
center
around
applying
the
fact
that
the
first
Chern
class
associated
to
the
diagonal
divisor
in
a
product
of
two
copies
of
a
proper
hyper-
bolic
curve
consists,
in
essence,
of
the
identity
matrix
[cf.
Lemma
1.3,
(iii)].
This
result
asserts,
roughly
speaking,
that
if
an
F-admissible
auto-
morphism
of
a
configuration
space
group
arises
from
an
F-admissible
automorphism
of
a
configuration
space
group
[arising
from
a
configu-
ration
space]
of
strictly
higher
dimension,
then
it
is
necessarily
FC-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)],
i.e.,
preserves
the
cuspidal
inertia
subgroups
of
the
various
subquotients
corresponding
to
surface
groups.
Theorem
A
(F-admissibility
and
FC-admissibility).
Let
Σ
be
a
set
of
prime
numbers
which
is
either
of
cardinality
one
or
equal
to
the
set
of
all
prime
numbers;
n
a
positive
integer;
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
X
a
hyperbolic
curve
of
type
(g,
r)
over
an
algebraically
closed
field
k
of
characteristic
∈
Σ;
X
n
the
n-th
configuration
space
of
X;
Π
n
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
X
n
;
“Out
FC
(−)”,
“Out
F
(−)”
⊆
“Out(−)”
the
subgroups
of
FC-
and
F-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
outomorphisms
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
of
“(−)”.
Then
the
following
hold:
(i)
Let
α
∈
Out
F
(Π
n+1
).
Then
α
induces
the
same
outomor-
phism
of
Π
n
relative
to
the
various
quotients
Π
n+1
Π
n
by
fiber
subgroups
of
length
1
[cf.
[MT],
Definition
2.3,
(iii)].
In
particular,
we
obtain
a
natural
homomorphism
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
.
(ii)
The
image
of
the
homomorphism
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
of
(i)
is
contained
in
Out
FC
(Π
n
)
⊆
Out
F
(Π
n
)
.
For
the
convenience
of
the
reader,
we
remark
that
our
treatment
of
Theorem
A
in
§1
does
not
require
any
knowledge
of
the
theory
of
semi-
graphs
of
anabelioids.
On
the
other
hand,
in
a
sequel
to
the
present
pa-
per,
we
intend
to
prove
a
substantial
stengthening
of
Theorem
A,
whose
Combinatorial
anabelian
topics
I
5
proof
makes
quite
essential
use
of
the
theory
of
[CmbGC],
[CmbCsp],
and
[NodNon]
[i.e.,
in
particular,
of
the
theory
of
semi-graphs
of
anabelioids
of
PSC-type].
In
§2
and
§3,
we
develop
various
technical
tools
that
will
play
a
cru-
cial
role
in
the
subsequent
development
of
the
theory
of
the
present
pa-
per.
In
§2,
we
study
various
fundamental
operations
on
semi-graphs
of
anabelioids
of
PSC-type.
A
more
detailed
description
of
these
opera-
tions
may
be
found
in
the
discussion
at
the
beginning
of
§2,
as
well
as
in
the
various
illustrations
referred
to
in
this
discussion.
Roughly
speaking,
these
operations
may
be
thought
of
as
abstract
profinite
combinatorial
analogues
of
various
well-known
operations
that
occur
in
the
theory
of
“surgery”
on
topological
surfaces
—
i.e.,
•
restriction
to
a
subsurface
arising
from
a
decomposition,
such
as
a
“pants
decomposition”,
of
the
surface
or
to
a
[suitably
positioned]
cycle;
•
partially
compactifying
the
surface
by
adding
“missing
points”;
•
cutting
a
surface
along
a
[suitably
positioned]
cycle;
•
gluing
together
two
surfaces
along
[suitably
positioned]
cy-
cles.
Most
of
§2
is
devoted
to
the
abstract
combinatorial
formulation
of
these
operations,
as
well
as
to
the
verification
of
various
basic
properties
in-
volving
these
operations.
In
§3,
we
develop
the
local
theory
of
the
second
cohomology
group
with
compact
supports
associated
to
various
sub-semi-graphs
and
com-
ponents
of
a
semi-graph
of
anabelioids
of
PSC-type.
Roughly
speaking,
this
theory
may
be
thought
of
as
a
sort
of
abstract
profinite
combina-
torial
analogue
of
the
local
theory
of
orientations
on
a
topological
surface
S,
i.e.,
the
theory
of
the
locally
defined
cohomology
modules
(U,
x)
→
H
2
(U,
U
\
{x};
Z)
(
∼
=
Z)
—
where
U
⊆
S
is
an
open
subset,
x
∈
U
.
In
the
abstract
profinite
combinatorial
context
of
the
present
paper,
the
various
locally
defined
second
cohomology
groups
with
compact
supports
give
rise
to
cyclo-
tomes,
i.e.,
copies
of
quotients
of
the
once-Tate-twisted
Galois
module
Z(1).
The
main
result
that
we
obtain
in
§3
concerns
various
canoni-
cal
synchronizations
of
cyclotomes
[cf.
Corollary
3.9],
i.e.,
canon-
ical
isomorphisms
between
these
cyclotomes
associated
to
various
local
6
Yuichiro
Hoshi
and
Shinichi
Mochizuki
portions
of
the
given
semi-graph
of
anabelioids
of
PSC-type
which
are
compatible
with
the
various
fundamental
operations
studied
in
§2.
In
§4,
we
apply
the
technical
tools
developed
in
§2,
§3
to
define
and
study
the
notion
of
a
profinite
Dehn
[multi-]twist
[cf.
Defini-
tion
4.4;
Theorem
4.8,
(iv)].
This
notion
is,
needless
to
say,
a
natural
abstract
profinite
combinatorial
analogue
of
the
usual
notion
of
a
Dehn
twist
in
the
theory
of
topological
surfaces.
On
the
other
hand,
it
is
de-
fined,
in
keeping
with
the
spirit
of
the
present
paper,
in
a
fashion
that
is
purely
combinatorial,
i.e.,
without
resorting
to
the
“crutch”
of
consid-
ering,
for
instance,
profinite
closures
of
Dehn
twists
associated
to
cycles
on
topological
surfaces.
Our
main
results
in
§4
[cf.
Theorem
4.8,
(i),
(iv);
Proposition
4.10,
(ii)]
assert,
roughly
speaking,
that
profinite
Dehn
twists
satisfy
a
structure
theory
of
the
sort
that
one
would
expect
from
the
analogy
with
the
topological
case,
and
that
this
structure
theory
is
compatible,
in
a
suitable
sense,
with
the
various
fundamental
operations
studied
in
§2.
Theorem
B
(Properties
of
profinite
Dehn
multi-twists).
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Write
Aut
|grph|
(G)
⊆
Aut(G)
for
the
group
of
automorphisms
of
G
which
induce
the
identity
automor-
phism
on
the
underlying
semi-graph
of
G
and
Dehn(G)
=
{
α
∈
Aut
|grph|
(G)
|
α
G|
v
=
id
G|
v
for
any
v
∈
Vert(G)
}
def
—
where
we
write
α
G|
v
for
the
restriction
of
α
to
the
semi-graph
of
anabelioids
G|
v
of
pro-Σ
PSC-type
determined
by
v
∈
Vert(G)
[cf.
Def-
initions
2.1,
(iii);
2.14,
(ii);
Remark
2.5.1,
(ii)];
we
shall
refer
to
an
element
of
Dehn(G)
as
a
profinite
Dehn
multi-twist
of
G.
Then
the
following
hold:
(i)
(Normality)
Dehn(G)
is
normal
in
Aut(G).
(ii)
(Structure
of
the
group
of
profinite
Dehn
multi-twists)
Write
def
Σ
),
Z
Σ
)
Λ
G
=
Hom
Z
Σ
(H
c
2
(G,
Z
for
the
cyclotome
associated
to
G
[cf.
Definitions
3.1,
(ii),
(iv);
3.8,
(i)].
Then
there
exists
a
natural
isomorphism
∼
Λ
G
D
G
:
Dehn(G)
−→
Node(G)
Combinatorial
anabelian
topics
I
7
that
is
functorial,
in
G,
with
respect
to
isomorphisms
of
semi-
graphs
of
anabelioids.
In
particular,
Dehn(G)
is
a
finitely
Σ
-module
of
rank
Node(G)
.
We
shall
generated
free
Z
refer
to
a
nontrivial
profinite
Dehn
multi-twist
whose
image
∈
Node(G)
Λ
G
lies
in
a
direct
summand
[i.e.,
in
a
single
“Λ
G
”]
as
a
profinite
Dehn
twist.
(iii)
(Exact
sequence
relating
profinite
Dehn
multi-twists
and
glueable
outomorphisms)
Write
Glu(G)
⊆
Aut
|grph|
(G|
v
)
v∈Vert(G)
for
the
[closed]
subgroup
of
“glueable”
collections
of
outomor-
phisms
of
the
direct
product
v∈Vert(G)
Aut
|grph|
(G|
v
)
consist-
ing
of
elements
(α
v
)
v∈Vert(G)
such
that
χ
v
(α
v
)
=
χ
w
(α
w
)
for
any
v,
w
∈
Vert(G)
—
where
we
write
G|
v
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
determined
by
v
∈
Vert(G)
[cf.
Σ
)
∗
for
the
pro-
Definition
2.1,
(iii)]
and
χ
v
:
Aut(G|
v
)
→
(
Z
Σ
cyclotomic
character
of
v
∈
Vert(G)
[cf.
Definition
3.8,
(ii)].
Then
the
natural
homomorphism
|grph|
(G|
v
)
Aut
|grph|
(G)
−→
v∈Vert(G)
Aut
α
→
(α
G|
v
)
v∈Vert(G)
factors
through
Glu(G)
⊆
v∈Vert(G)
Aut
|grph|
(G|
v
),
and,
more-
over,
the
resulting
homomorphism
ρ
Vert
:
Aut
|grph|
(G)
→
G
Glu(G)
[cf.
(i)]
fits
into
an
exact
sequence
of
profinite
groups
ρ
Vert
G
1
−→
Dehn(G)
−→
Aut
|grph|
(G)
−→
Glu(G)
−→
1
.
The
approach
of
§2,
§3,
§4
is
purely
combinatorial
in
nature.
On
the
other
hand,
in
§5,
we
return
briefly
to
the
world
of
[log]
schemes
in
or-
der
to
compare
the
purely
combinatorial
constructions
of
§2,
§3,
§4
to
analogous
constructions
from
scheme
theory.
The
main
techinical
result
[cf.
Theorem
5.7]
of
§5
asserts
that
the
purely
combinatorial
synchronizations
of
cyclotomes
constructed
in
§3,
§4
for
the
profinite
Dehn
twists
associated
to
the
various
nodes
of
the
semi-graph
of
anabe-
lioids
of
PSC-type
under
consideration
coincide
with
certain
natural
scheme-theoretic
synchronizations
of
cyclotomes.
This
technical
result
is
obtained,
roughly
speaking,
by
applying
the
various
fundamen-
tal
operations
of
§2
so
as
to
reduce
to
the
case
where
the
semi-graph
of
8
Yuichiro
Hoshi
and
Shinichi
Mochizuki
anabelioids
of
PSC-type
under
consideration
admits
a
symmetry
that
permutes
the
nodes
[cf.
Fig.
6];
the
desired
coincidence
of
synchro-
nizations
is
then
obtained
by
observing
that
both
the
combinatorial
and
the
scheme-theoretic
synchronizations
are
compatible
with
this
symme-
try.
One
way
to
understand
this
fundamental
coincidence
of
synchro-
nizations
is
as
a
sort
of
abstract
combinatorial
analogue
of
the
cyclotomic
synchronization
given
in
[GalSct],
Theorem
4.3;
[AbsHyp],
Lemma
2.5,
(ii)
[cf.
Remark
5.9.1,
(i)].
Another
way
to
understand
this
fundamental
coincidence
of
synchronizations
is
as
a
statement
to
the
effect
that
the
Galois
action
associated
to
a
[scheme-theoretic!]
degenerating
family
of
hyperbolic
curves
over
a
com-
plete
equicharacteristic
discrete
valuation
ring
of
characteristic
zero
—
i.e.,
“an
outer
representation
of
IPSC-type”
—
admits
a
purely
combinatorial
description
[cf.
Corollary
5.9,
(iii)].
That
is
to
say,
one
central
problem
in
the
theory
of
outer
Galois
repre-
sentations
associated
to
hyperbolic
curves
over
arithmetic
fields
is
pre-
cisely
the
problem
of
giving
such
a
“purely
combinatorial
description”
of
the
outer
Galois
representation.
Indeed,
this
point
of
view
plays
a
central
role
in
the
theory
of
the
Grothendieck-Teichmüller
group.
Thus,
although
an
explicit
solution
to
this
problem
is
well
out
of
reach
at
the
present
time
in
the
case
of
number
fields
or
mixed-characteristic
lo-
cal
fields,
the
theory
of
§5
yields
a
solution
to
this
problem
in
the
case
of
complete
equicharacteristic
discrete
valuation
fields
of
characteristic
zero.
One
consequence
of
this
solution
is
the
following
criterion
for
an
outer
representation
to
be
of
IPSC-type
[cf.
Corollary
5.10].
Theorem
C
(Combinatorial/group-theoretic
nature
of
scheme-theoreticity).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
Σ
a
nonempty
set
of
prime
numbers;
R
a
complete
discrete
valuation
ring
whose
residue
field
k
is
separably
closed
of
char-
def
acteristic
∈
Σ;
S
log
the
log
scheme
obtained
by
equipping
S
=
Spec
R
with
the
log
structure
determined
by
the
maximal
ideal
of
R;
(M
g,r
)
S
the
moduli
stack
of
r-pointed
stable
curves
of
genus
g
over
S
whose
r
marked
points
are
equipped
with
an
ordering;
(M
g,r
)
S
⊆
(M
g,r
)
S
log
the
open
substack
of
(M
g,r
)
S
parametrizing
smooth
curves;
(M
g,r
)
S
the
log
stack
obtained
by
equipping
(M
g,r
)
S
with
the
log
structure
associ-
ated
to
the
divisor
with
normal
crossings
(M
g,r
)
S
\
(M
g,r
)
S
⊆
(M
g,r
)
S
;
the
completion
of
the
x
∈
(M
g,r
)
S
(k)
a
k-valued
point
of
(M
g,r
)
S
;
O
log
the
log
scheme
obtained
by
local
ring
of
(M
g,r
)
S
at
the
image
of
x;
T
Combinatorial
anabelian
topics
I
9
with
the
log
structure
induced
by
the
log
struc-
equipping
T
=
Spec
O
log
log
the
log
scheme
obtained
by
equipping
the
closed
ture
of
(M
g,r
)
S
;
t
point
of
T
with
the
log
structure
induced
by
the
log
structure
of
T
log
;
X
t
log
the
stable
log
curve
over
t
log
corresponding
to
the
natural
strict
(1-
log
)morphism
t
log
→
(M
g,r
)
S
;
I
T
log
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
π
1
(T
log
)
of
T
log
;
I
S
log
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
π
1
(S
log
)
of
S
log
;
G
X
log
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
determined
by
the
stable
log
curve
X
t
log
[cf.
[CmbGC],
Example
2.5];
ρ
univ
:
I
T
log
→
Aut(G
X
log
)
the
natural
X
log
def
t
outer
representation
associated
to
X
t
log
[cf.
Definition
5.5];
I
a
profinite
group;
ρ
:
I
→
Aut(G
X
log
)
an
outer
representation
of
pro-Σ
PSC-type
[cf.
[NodNon],
Definition
2.1,
(i)].
Then
the
following
conditions
are
equivalent:
(i)
ρ
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)].
(ii)
There
exist
a
morphism
of
log
schemes
φ
log
:
S
log
→
T
log
over
S
and
an
isomorphism
of
outer
representations
of
pro-Σ
∼
◦
I
φ
log
[cf.
[NodNon],
Definition
2.1,
(i)]
PSC-type
ρ
→
ρ
univ
X
t
log
—
where
we
write
I
φ
log
:
I
S
log
→
I
T
log
for
the
homomorphism
induced
by
φ
log
—
i.e.,
there
exist
an
automorphism
β
of
∼
G
X
log
and
an
isomorphism
α
:
I
→
I
S
log
such
that
the
diagram
ρ
−−−−→
I
⏐
⏐
α
ρ
X
Aut(G
X
log
)
⏐
⏐
log
◦I
φ
log
t
I
S
log
−−−
−−−−→
Aut(G
X
log
)
—
where
the
right-hand
vertical
arrow
is
the
automorphism
of
Aut(G
X
log
)
induced
by
β
—
commutes.
(iii)
There
exist
a
morphism
of
log
schemes
φ
log
:
S
log
→
T
log
over
∼
S
and
an
isomorphism
α
:
I
→
I
S
log
such
that
ρ
=
ρ
univ
◦I
φ
log
◦
X
t
log
α
—
where
we
write
I
φ
log
:
I
S
log
→
I
T
log
for
the
homomorphism
induced
by
φ
log
—
i.e.,
the
automorphism
“β”
of
(ii)
may
be
taken
to
be
the
identity.
Before
proceeding,
in
this
context
we
observe
that
one
fundamen-
tal
intrinsic
difference
between
outer
representations
of
IPSC-type
and
more
general
outer
representations
of
NN-type
is
that,
unlike
the
case
with
outer
representations
of
IPSC-type,
the
period
matrices
associated
10
Yuichiro
Hoshi
and
Shinichi
Mochizuki
to
outer
representations
of
NN-type
may,
in
general,
fail
to
be
nonde-
generate
—
cf.
the
discussion
of
Remark
5.9.2.
Here,
we
remark
in
passing
that
in
a
sequel
to
the
present
paper,
the
theory
of
§5
will
play
an
important
role
in
the
proofs
of
certain
applications
to
the
theory
of
tempered
fundamental
groups
developed
in
[André].
Finally,
in
§6,
we
apply
the
theory
of
profinite
Dehn
twists
developed
in
§4
to
prove
a
“geometric
version
of
the
Grothendieck
Conjec-
ture”
for
—
i.e.,
put
another
way,
we
compute
the
centralizer
of
the
geometric
monodromy
associated
to
—
the
tautological
curve
over
the
moduli
stack
of
pointed
smooth
curves
[cf.
Theorems
6.13;
6.14].
Theorem
D
(Centralizers
of
geometric
monodromy
groups
arising
from
moduli
stacks
of
pointed
curves).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
Σ
a
nonempty
set
of
prime
numbers;
k
an
algebraically
closed
field
of
characteristic
zero.
Write
(M
g,r
)
k
for
the
moduli
stack
of
r-pointed
smooth
curves
of
genus
g
over
k
whose
r
marked
points
are
equipped
with
an
order-
ing;
C
g,r
→
M
g,r
for
the
tautological
curve
over
M
g,r
[cf.
the
dis-
def
cussion
entitled
“Curves”
in
§0];
Π
M
g,r
=
π
1
((M
g,r
)
k
)
for
the
étale
fundamental
group
of
the
moduli
stack
(M
g,r
)
k
;
Π
g,r
for
the
maximal
pro-Σ
quotient
of
the
kernel
N
g,r
of
the
natural
surjection
π
1
((C
g,r
)
k
)
π
1
((M
g,r
)
k
)
=
Π
M
g,r
;
Π
C
g,r
for
the
quotient
of
the
étale
fundamen-
tal
group
π
1
((C
g,r
)
k
)
of
(C
g,r
)
k
by
the
kernel
of
the
natural
surjection
N
g,r
Π
g,r
;
Out
C
(Π
g,r
)
for
the
group
of
outomorphisms
[cf.
the
dis-
cussion
entitled
“Topological
groups”
in
§0]
of
Π
g,r
which
induce
bijec-
tions
on
the
set
of
cuspidal
inertia
subgroups
of
Π
g,r
.
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
g,r
−→
Π
C
g,r
−→
Π
M
g,r
−→
1
,
which
determines
an
outer
representation
ρ
g,r
:
Π
M
g,r
−→
Out(Π
g,r
)
.
Then
the
following
hold:
(i)
Let
H
⊆
Π
M
g,r
be
an
open
subgroup
of
Π
M
g,r
.
Suppose
that
one
of
the
following
two
conditions
is
satisfied:
(a)
2g
−
2
+
r
>
1,
i.e.,
(g,
r)
∈
{(0,
3),
(1,
1)}.
(b)
(g,
r)
=
(1,
1),
2
∈
Σ,
and
H
=
Π
M
g,r
.
Combinatorial
anabelian
topics
I
11
Then
the
composite
of
natural
homomorphisms
Aut
(M
g,r
)
k
((C
g,r
)
k
)
−→
Aut
Π
M
g,r
(Π
C
g,r
)/Inn(Π
g,r
)
∼
−→
Z
Out(Π
g,r
)
(Im(ρ
g,r
))
⊆
Z
Out(Π
g,r
)
(ρ
g,r
(H))
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
deter-
mines
an
isomorphism
∼
Aut
(M
g,r
)
k
((C
g,r
)
k
)
−→
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H))
.
Here,
we
recall
that
Aut
(M
g,r
)
k
((C
g,r
)
k
)
is
isomorphic
to
⎧
if
(g,
r)
=
(0,
4);
⎨
Z/2Z
×
Z/2Z
Z/2Z
if
(g,
r)
∈
{(1,
1),
(1,
2),
(2,
0)};
⎩
{1}
if
(g,
r)
∈
{(0,
4),
(1,
1),
(1,
2),
(2,
0)}
.
(ii)
§0.
Let
H
⊆
Out
C
(Π
g,r
)
be
a
closed
subgroup
of
Out
C
(Π
g,r
)
that
contains
an
open
subgroup
of
Im(ρ
g,r
)
⊆
Out(Π
g,r
).
Suppose
that
2g
−
2
+
r
>
1,
i.e.,
(g,
r)
∈
{(0,
3),
(1,
1)}.
Then
H
is
almost
slim
[cf.
the
discussion
entitled
“Topolog-
ical
groups”
in
§0].
If,
moreover,
2g
−
2
+
r
>
2,
i.e.,
(g,
r)
∈
{(0,
3),
(0,
4),
(1,
1),
(1,
2),
(2,
0)},
then
H
is
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Notations
and
Conventions
Sets:
If
S
is
a
set,
then
we
shall
denote
by
2
S
the
power
set
of
S
and
by
S
the
cardinality
of
S.
Numbers:
The
notation
Primes
will
be
used
to
denote
the
set
of
all
prime
numbers.
The
notation
N
will
be
used
to
denote
the
set
or
[ad-
ditive]
monoid
of
nonnegative
rational
integers.
The
notation
Z
will
be
used
to
denote
the
set,
group,
or
ring
of
rational
integers.
The
notation
Q
will
be
used
to
denote
the
set,
group,
or
field
of
rational
numbers.
will
be
used
to
denote
the
profinite
completion
of
Z.
The
notation
Z
If
p
∈
Primes,
then
the
notation
Z
p
(respectively,
Q
p
)
will
be
used
to
denote
the
p-adic
completion
of
Z
(respectively,
Q).
If
Σ
⊆
Primes,
then
Σ
will
be
used
to
denote
the
pro-Σ
completion
of
Z.
the
notation
Z
12
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Monoids:
We
shall
write
M
gp
for
the
groupification
of
a
monoid
M
.
Topological
groups:
Let
G
be
a
topological
group
and
P
a
property
of
topological
groups
[e.g.,
“abelian”
or
“pro-Σ”
for
some
Σ
⊆
Primes].
Then
we
shall
say
that
G
is
almost
P
if
there
exists
an
open
subgroup
of
G
that
is
P.
Let
G
be
a
topological
group
and
H
⊆
G
a
closed
subgroup
of
G.
Then
we
shall
denote
by
Z
G
(H)
(respectively,
N
G
(H);
C
G
(H))
the
centralizer
(respectively,
normalizer;
commensurator)
of
H
in
G,
i.e.,
Z
G
(H)
=
{
g
∈
G
|
ghg
−1
=
h
for
any
h
∈
H
}
,
def
(respectively,
N
G
(H)
=
{
g
∈
G
|
g
·
H
·
g
−1
=
H
}
;
def
C
G
(H)
=
{
g
∈
G
|
H
∩
g·H·g
−1
is
of
finite
index
in
H
and
g·H·g
−1
}
);
def
def
we
shall
refer
to
Z(G)
=
Z
G
(G)
as
the
center
of
G.
It
is
immediate
from
the
definitions
that
Z
G
(H)
⊆
N
G
(H)
⊆
C
G
(H)
;
H
⊆
N
G
(H)
.
We
shall
say
that
the
closed
subgroup
H
is
centrally
terminal
(respec-
tively,
normally
terminal;
commensurably
terminal)
in
G
if
H
=
Z
G
(H)
(respectively,
H
=
N
G
(H);
H
=
C
G
(H)).
We
shall
say
that
G
is
slim
if
Z
G
(U
)
=
{1}
for
any
open
subgroup
U
of
G.
Let
G
be
a
topological
group.
Then
we
shall
write
G
ab
for
the
abelianization
of
G,
i.e.,
the
quotient
of
G
by
the
closure
of
the
commu-
tator
subgroup
of
G.
Let
G
be
a
topological
group.
Then
we
shall
write
Aut(G)
for
the
group
of
[continuous]
automorphisms
of
G,
Inn(G)
⊆
Aut(G)
for
the
def
group
of
inner
automorphisms
of
G,
and
Out(G)
=
Aut(G)/Inn(G).
We
shall
refer
to
an
element
of
Out(G)
as
an
outomorphism
of
G.
Now
suppose
that
G
is
center-free
[i.e.,
Z(G)
=
{1}].
Then
we
have
an
exact
sequence
of
groups
∼
1
−→
G
(
→
Inn(G))
−→
Aut(G)
−→
Out(G)
−→
1
.
If
J
is
a
group
and
ρ
:
J
→
Out(G)
is
a
homomorphism,
then
we
shall
denote
by
out
G
J
the
group
obtained
by
pulling
back
the
above
exact
sequence
of
profinite
groups
via
ρ.
Thus,
we
have
a
natural
exact
sequence
of
groups
out
1
−→
G
−→
G
J
−→
J
−→
1
.
Combinatorial
anabelian
topics
I
13
Suppose
further
that
G
is
profinite
and
topologically
finitely
generated.
Then
one
verifies
easily
that
the
topology
of
G
admits
a
basis
of
char-
acteristic
open
subgroups,
which
thus
induces
a
profinite
topology
on
the
groups
Aut(G)
and
Out(G)
with
respect
to
which
the
above
exact
se-
quence
relating
Aut(G)
and
Out(G)
determines
an
exact
sequence
of
profinite
groups.
In
particular,
one
verifies
easily
that
if,
moreover,
J
is
profinite
and
ρ
:
J
→
Out(G)
is
continuous,
then
the
above
exact
out
sequence
involving
G
J
determines
an
exact
sequence
of
profinite
groups.
Let
G,
J
be
profinite
groups.
Suppose
that
G
is
center-free
and
topologically
finitely
generated.
Let
ρ
:
J
→
Out(G)
be
a
continuous
out
homomorphism.
Write
Aut
J
(G
J)
for
the
group
of
[continuous]
out
automorphisms
of
G
J
that
preserve
and
induce
the
identity
auto-
morphism
on
the
quotient
J.
Then
one
verifies
easily
that
the
operation
of
restricting
to
G
determines
an
isomorphism
of
profinite
groups
out
∼
Aut
J
(G
J)/Inn(G)
−→
Z
Out(G)
(Im(ρ))
.
Let
G
and
H
be
topological
groups.
Then
we
shall
refer
to
a
homo-
morphism
of
topological
groups
φ
:
G
→
H
as
a
split
injection
(respec-
tively,
split
surjection)
if
there
exists
a
homomorphism
of
topological
groups
ψ
:
H
→
G
such
that
ψ
◦
φ
(respectively,
φ
◦
ψ)
is
the
identity
automorphism
of
G
(respectively,
H).
Log
schemes:
When
a
scheme
appears
in
a
diagram
of
log
schemes,
the
scheme
is
to
be
understood
as
the
log
scheme
obtained
by
equipping
the
scheme
with
the
trivial
log
structure.
If
X
log
is
a
log
scheme,
then
we
shall
refer
to
the
largest
open
subscheme
of
the
underlying
scheme
of
X
log
over
which
the
log
structure
is
trivial
as
the
interior
of
X
log
.
Fiber
products
of
fs
log
schemes
are
to
be
understood
as
fiber
products
taken
in
the
category
of
fs
log
schemes.
Curves:
We
shall
use
the
terms
“hyperbolic
curve”,
“cusp”,
“stable
log
curve”,
“smooth
log
curve”,
and
“tripod”
as
they
are
defined
in
[CmbGC],
§0;
[Hsh],
§0.
If
(g,
r)
is
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0,
then
we
shall
denote
by
M
g,r
the
moduli
stack
of
r-pointed
stable
curves
of
genus
g
over
Z
whose
r
marked
points
are
equipped
with
an
ordering,
by
M
g,r
⊆
M
g,r
the
open
substack
of
M
g,r
parametrizing
log
smooth
curves,
by
M
g,r
the
log
stack
obtained
by
equipping
M
g,r
with
the
log
structure
associated
to
the
divisor
with
normal
crossings
M
g,r
\
14
Yuichiro
Hoshi
and
Shinichi
Mochizuki
M
g,r
⊆
M
g,r
,
by
C
g,r
→
M
g,r
the
tautological
curve
over
M
g,r
,
and
by
D
g,r
⊆
C
g,r
the
corresponding
tautological
divisor
of
marked
points
of
C
g,r
→
M
g,r
.
Then
the
divisor
given
by
the
union
of
D
g,r
with
the
inverse
image
in
C
g,r
of
the
divisor
M
g,r
\
M
g,r
⊆
M
g,r
determines
log
a
log
structure
on
C
g,r
;
denote
the
resulting
log
stack
by
C
g,r
.
Thus,
log
log
we
obtain
a
(1-)morphism
of
log
stacks
C
g,r
→
M
g,r
.
We
shall
denote
log
by
C
g,r
⊆
C
g,r
the
interior
of
C
g,r
.
Thus,
we
obtain
a
(1-)morphism
of
def
stacks
C
g,r
→
M
g,r
.
Let
S
be
a
scheme.
Then
we
shall
write
(M
g,r
)
S
=
log
log
def
def
M
g,r
×
Spec
Z
S,
(M
g,r
)
S
=
M
g,r
×
Spec
Z
S,
(M
g,r
)
S
=
M
g,r
×
Spec
Z
log
def
def
def
S,
(C
g,r
)
S
=
C
g,r
×
Spec
Z
S,
(C
g,r
)
S
=
C
g,r
×
Spec
Z
S,
and
(C
g,r
)
S
=
log
C
g,r
×
Spec
Z
S.
log
Let
n
be
a
positive
integer
and
X
a
stable
log
curve
of
type
(g,
r)
over
a
log
scheme
S
log
.
Then
we
shall
refer
to
the
log
scheme
obtained
log
log
by
pulling
back
the
(1-)morphism
M
g,r+n
→
M
g,r
given
by
forgetting
log
the
last
n
points
via
the
classifying
(1-)morphism
S
log
→
M
g,r
of
X
log
as
the
n-th
log
configuration
space
of
X
log
.
§1.
F-admissibility
and
FC-admissibility
In
the
present
§,
we
consider
the
FC-admissibility
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
of
F-admissible
automorphisms
[cf.
[CmbCsp],
Def-
inition
1.1,
(ii)]
of
configuration
space
groups
[cf.
[MT],
Definition
2.3,
(i)].
Roughly
speaking,
we
prove
that
if
an
F-admissible
automorphism
of
a
configuration
space
group
arises
from
an
F-admissible
automor-
phism
of
a
configuration
space
group
[arising
from
a
configuration
space]
of
strictly
higher
dimension,
then
it
is
necessarily
FC-admissible,
i.e.,
preserves
the
cuspidal
inertia
subgroups
of
the
various
subquotients
cor-
responding
to
surface
groups
[cf.
Theorem
1.8,
Corollary
1.9
below].
Lemma
1.1
(Representations
arising
from
certain
families
of
hyperbolic
curves).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
l
a
prime
number;
k
an
algebraically
closed
field
of
characteristic
=
l;
B
and
C
hyperbolic
curves
over
k
of
type
(g,
r);
n
a
positive
integer.
Suppose
that
(r,
n)
=
(0,
1).
For
i
=
1,
·
·
·
,
n,
let
pr
∼
f
i
:
B
→
C
be
an
isomorphism
over
k;
s
i
the
section
of
B
×
k
C
→
1
B
determined
by
the
isomorphism
f
i
.
Suppose
that,
for
any
i
=
j,
Combinatorial
anabelian
topics
I
15
Im(s
i
)
∩
Im(s
j
)
=
∅.
Write
def
Z
=
B
×
k
C
\
Im(s
i
)
⊆
B
×
k
C
i=1,···
,n
for
the
complement
of
the
images
of
the
s
i
’s,
where
i
ranges
over
the
pr
integers
such
that
1
≤
i
≤
n;
pr
for
the
composite
Z
→
B
×
k
C
→
1
B
[thus,
pr
:
Z
→
B
is
a
family
of
hyperbolic
curves
of
type
(g,
r
+
n)];
Π
B
(respectively,
Π
C
;
Π
Z
)
the
maximal
pro-l
quotient
of
the
étale
fundamental
group
π
1
(B)
(respectively,
π
1
(C);
π
1
(Z))
of
B
(respectively,
C;
Z);
pr
:
Π
Z
Π
B
for
the
surjection
induced
by
pr;
Π
Z/B
for
the
kernel
of
pr;
ρ
Z/B
:
Π
B
→
Out(Π
Z/B
)
for
the
outer
representation
of
Π
B
on
Π
Z/B
determined
by
the
exact
sequence
pr
1
−→
Π
Z/B
−→
Π
Z
−→
Π
B
−→
1
.
Let
b
be
a
geometric
point
of
B
and
Z
b
the
geometric
fiber
of
pr
:
Z
→
B
at
b.
For
i
=
1,
·
·
·
,
n,
fix
an
inertia
subgroup
[among
its
various
conjugates]
of
the
étale
fundamental
group
π
1
(Z
b
)
of
Z
b
associated
to
the
cusp
of
Z
b
determined
by
the
section
s
i
and
denote
by
I
s
i
⊆
Π
Z/B
the
image
in
Π
Z/B
of
this
inertia
subgroup
of
π
1
(Z
b
).
Then
the
following
hold:
(i)
(Fundamental
groups
of
fibers)
The
quotient
Π
Z/B
of
the
étale
fundamental
group
π
1
(Z
b
)
of
the
geometric
fiber
Z
b
co-
incides
with
the
maximal
pro-l
quotient
of
π
1
(Z
b
).
(ii)
(Abelianizations
of
the
fundamental
groups
of
fibers)
For
i
=
1,
·
·
·
,
n,
write
J
s
i
⊆
Π
ab
Z/B
for
the
image
of
I
s
i
⊆
Π
Z/B
ab
in
Π
ab
.
Then
the
composite
I
s
i
→
Π
Z/B
Π
Z/B
determines
Z/B
∼
an
isomorphism
I
s
i
→
J
s
i
;
moreover,
the
inclusions
J
s
i
→
Π
ab
Z/B
determine
an
exact
sequence
1
−→
(
n
ab
J
s
i
)/J
r
−→
Π
ab
Z/B
−→
Π
C
−→
1
i=1
—
where
J
r
⊆
n
i=1
J
s
i
16
Yuichiro
Hoshi
and
Shinichi
Mochizuki
is
a
Z
l
-submodule
such
that
J
r
≃
Z
l
0
if
r
=
0,
if
r
=
0,
and,
moreover,
if
r
=
0
and
i
=
1,
·
·
·
,
n,
then
the
composite
J
r
→
n
pr
si
J
s
i
J
s
i
i=1
is
an
isomorphism.
(iii)
(Unipotency
of
a
certain
natural
representation)
The
action
of
Π
B
on
Π
ab
Z/B
determined
by
ρ
Z/B
preserves
the
exact
sequence
1
−→
(
n
ab
J
s
i
)/J
r
−→
Π
ab
Z/B
−→
Π
C
−→
1
i=1
[cf.
(ii)]
and
induces
the
identity
automorphisms
on
the
n
subquotients
(
i=1
J
s
i
)/J
r
and
Π
ab
C
;
in
particular,
the
nat-
ural
homomorphism
Π
B
→
Aut
Z
l
(Π
ab
Z/B
)
factors
through
a
uniquely
determined
homomorphism
n
Π
B
−→
Hom
Z
l
Π
ab
,
(
J
)/J
.
s
r
C
i
i=1
Proof.
Assertion
(i)
follows
immediately
from
the
[easily
verified]
fact
that
the
natural
action
of
π
1
(B)
on
π
1
(Z
b
)
ab
⊗
Z
Z
l
is
unipotent
—
cf.,
e.g.,
[Hsh],
Proposition
1.4,
(i),
for
more
details.
[Note
that
although
[Hsh],
Proposition
1.4,
(i),
is
only
stated
in
the
case
where
the
hyperbolic
curves
corresponding
to
B
and
C
are
proper,
the
same
proof
may
be
applied
to
the
case
where
these
hyperbolic
curves
are
affine.]
Assertion
(ii)
follows
immediately,
in
light
of
our
assumption
that
(r,
n)
=
(0,
1),
from
assertion
(i),
together
with
the
well-known
structure
of
the
maximal
pro-l
quotient
of
the
fundamental
group
of
a
smooth
curve
over
an
algebraically
closed
field
of
characteristic
=
l.
Finally,
we
verify
assertion
(iii).
The
fact
that
the
action
of
Π
B
on
Π
Z/B
preserves
the
exact
sequence
appearing
in
the
statement
of
assertion
(iii)
ab
follows
immediately
from
the
fact
that
the
surjection
Π
ab
Z/B
Π
C
is
induced
by
the
open
immersion
Z
→
B
×
k
C
over
B.
The
fact
that
the
n
action
in
question
induces
the
identity
automorphism
on
(
i=1
J
s
i
)/J
r
Combinatorial
anabelian
topics
I
17
(respectively,
Π
ab
C
)
follows
immediately
from
the
fact
that
the
f
i
’s
are
ab
isomorphisms
(respectively,
the
fact
that
the
surjection
Π
ab
Z/B
Π
C
is
Q.E.D.
induced
by
the
open
immersion
Z
→
B
×
k
C
over
B).
Lemma
1.2
(Maximal
cuspidally
central
quotients
of
certain
fundamental
groups).
In
the
notation
of
Lemma
1.1,
for
i
=
1,
·
·
·
,
n,
write
Π
Z/B
Π
(Z/B)[i]
(
Π
C
)
for
the
quotient
of
Π
Z/B
by
the
normal
closed
subgroup
topologically
normally
generated
by
the
I
s
j
’s,
where
j
ranges
over
the
integers
such
that
1
≤
j
≤
n
and
j
=
i;
Π
(Z/B)[i]
E
(Z/B)[i]
for
the
maximal
cuspidally
central
quotient
[cf.
[AbsCsp],
Defini-
tion
1.1,
(i)]
relative
to
the
surjection
Π
(Z/B)[i]
Π
C
determined
by
the
natural
open
immersion
Z
→
B
×
k
C;
I
s
E
i
⊆
E
(Z/B)[i]
for
the
kernel
of
the
natural
surjection
E
(Z/B)[i]
Π
C
;
and
E
Z/B
def
=
E
(Z/B)[1]
×
Π
C
·
·
·
×
Π
C
E
(Z/B)[n]
.
Then
the
following
hold:
(i)
(Cuspidal
inertia
subgroups)
Let
1
≤
i,
j
≤
n
be
integers.
Then
the
homomorphism
I
s
i
→
I
s
E
j
determined
by
the
compos-
ite
I
s
i
→
Π
Z/B
E
(Z/B)[j]
is
an
isomorphism
(respectively,
trivial)
if
i
=
j
(respectively,
i
=
j).
(ii)
(Surjectivity)
The
homomorphism
Π
Z/B
→
E
Z/B
determined
by
the
natural
surjections
Π
Z/B
E
(Z/B)[i]
—
where
i
ranges
over
the
integers
such
that
1
≤
i
≤
n
—
is
surjective.
(iii)
(Maximal
cuspidally
central
quotients
and
abelianiza-
tions)
The
quotient
Π
Z/B
E
Z/B
of
Π
Z/B
[cf.
(ii)]
coin-
cides
with
the
maximal
cuspidally
central
quotient
[cf.
[AbsCsp],
Definition
1.1,
(i)]
relative
to
the
surjection
Π
Z/B
Π
C
determined
by
the
natural
open
immersion
Z
→
B
×
k
C.
In
particular,
the
natural
surjection
Π
Z/B
Π
ab
Z/B
factors
18
Yuichiro
Hoshi
and
Shinichi
Mochizuki
through
the
surjection
Π
Z/B
E
Z/B
,
and
the
resulting
sur-
jection
E
Z/B
Π
ab
Z/B
fits
into
a
commutative
diagram
n
E
−−−−→
E
Z/B
−−−−→
Π
C
−−−−→
1
1
−−−−→
i=1
I
s
i
⏐
⏐
⏐
⏐
⏐
⏐
n
ab
1
−−−−→
(
i=1
J
s
i
)/J
r
−−−−→
Π
ab
Z/B
−−−−→
Π
C
−−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
surjective.
Moreover,
the
left-hand
vertical
arrow
coincides
with
the
surjection
induced
by
the
natural
isomor-
∼
∼
phisms
I
s
i
→
J
s
i
[cf.
Lemma
1.1,
(ii)]
and
I
s
i
→
I
s
E
i
[cf.
(i)].
Finally,
if
r
=
0,
then
the
right-hand
square
is
cartesian.
Proof.
Assertion
(i)
follows
immediately
from
the
definition
of
the
quotient
E
(Z/B)[j]
of
Π
Z/B
,
together
with
the
well-known
structure
of
the
maximal
pro-l
quotient
of
the
fundamental
group
of
a
smooth
curve
over
an
algebraically
closed
field
of
characteristic
=
l
[cf.,
e.g.,
[MT],
Lemma
4.2,
(iv),
(v)].
Assertion
(ii)
follows
immediately
from
assertion
(i).
Assertion
(iii)
follows
immediately
from
assertions
(i),
(ii)
[cf.
[AbsCsp],
Proposition
1.6,
(iii)].
Q.E.D.
Lemma
1.3
(The
kernels
of
representations
arising
from
cer-
tain
families
of
hyperbolic
curves).
In
the
notation
of
Lemmas
1.1,
1.2,
suppose
that
r
=
0.
Then
the
following
hold:
(i)
(Unipotency
of
a
certain
natural
outer
representation)
Consider
the
action
of
Π
B
on
E
Z/B
determined
by
the
natural
isomorphism
∼
Π
C
E
Z/B
−→
Π
ab
Z/B
×
Π
ab
C
[cf.
Lemma
1.2,
(iii)],
together
with
the
natural
action
of
Π
B
on
Π
ab
Z/B
induced
by
ρ
Z/B
and
the
trivial
action
of
Π
B
on
Π
C
.
Then
the
outer
action
of
Π
B
on
E
Z/B
induced
by
this
ac-
tion
coincides
with
the
natural
outer
action
of
Π
B
on
E
Z/B
induced
by
ρ
Z/B
.
In
particular,
relative
to
the
natural
identifi-
∼
cation
I
s
i
→
I
s
E
i
[cf.
Lemma
1.2,
(i)],
the
above
action
of
Π
B
on
E
Z/B
factors
through
the
homomorphism
n
n
∼
Π
B
−→
Hom
Z
l
Π
C
,
I
s
i
−→
Hom
Z
l
Π
ab
I
s
i
C
,
i=1
i=1
Combinatorial
anabelian
topics
I
19
obtained
in
Lemma
1.1,
(iii).
(ii)
(Homomorphisms
arising
from
a
certain
extension)
For
i
=
1,
·
·
·
,
n,
write
φ
i
for
the
composite
n
ab
,
I
,
I
Π
B
−→
Hom
Z
l
Π
ab
−→
Hom
Π
s
Z
s
C
C
j
i
l
j=1
—
where
the
first
arrow
is
the
homomorphism
of
(i),
and
the
second
arrow
n
is
the
homomorphism
determined
by
the
projec-
tion
pr
i
:
j=1
I
s
j
I
s
i
.
Then
the
homomorphism
φ
i
co-
incides
with
the
image
of
the
element
of
H
2
(Π
B
×
Π
C
,
I
s
i
)
determined
by
the
extension
1
−→
I
s
i
−→
Π
E
Z[i]
−→
Π
B
×
Π
C
−→
1
def
—
where
we
write
Π
E
Z[i]
=
Π
Z
/Ker(Π
Z/B
E
Z/B[i]
)
—
of
∼
Π
B
×
Π
C
by
I
s
i
→
I
s
E
i
[cf.
Lemma
1.2,
(i)]
via
the
composite
∼
∼
H
2
(Π
B
×
Π
C
,
I
s
i
)
→
H
1
(Π
B
,
H
1
(Π
C
,
I
s
i
))
→
Hom
Π
B
,
Hom(Π
C
,
I
s
i
)
—
where
the
first
arrow
is
the
isomorphism
determined
by
the
Hochschild-Serre
spectral
sequence
relative
to
the
surjection
pr
1
Π
B
×
Π
C
Π
B
.
(iii)
(Factorization)
Write
B
(respectively,
C)
for
the
compactifi-
cation
of
C
(respectively,
B)
and
Π
B
(respectively,
Π
C
)
for
the
maximal
pro-l
quotient
of
the
étale
fundamental
group
π
1
(B)
(respectively,
π
1
(C))
of
B
(respectively,
C).
Then
the
homo-
morphism
φ
i
of
(ii)
factors
as
the
composite
∼
∼
→
Π
ab
→
Hom
Z
l
Π
ab
,
I
s
i
→
Hom
Z
l
Π
ab
Π
B
Π
ab
C
,
I
s
i
B
C
C
—
where
the
first
(respectively,
second;
fourth)
arrow
is
the
∼
homomorphism
induced
by
B
→
B
(respectively,
f
i
:
B
→
C;
C
→
C),
and
the
third
arrow
is
the
isomorphism
determined
by
the
Poincaré
duality
isomorphism
in
étale
cohomology,
rel-
∼
ative
to
the
natural
isomorphism
I
s
i
→
Z
l
(1).
[Here,
the
“(1)”
denotes
a
“Tate
twist”.]
(iv)
(Kernel
of
a
certain
natural
representation)
The
kernel
of
the
homomorphism
Π
B
→
Aut
Z
l
(Π
ab
Z/B
)
determined
by
ρ
Z/B
coincides
with
the
kernel
of
the
natural
surjection
Π
B
Π
ab
.
B
20
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
various
defi-
nitions
involved.
Next,
we
verify
assertion
(iii).
It
follows
from
assertion
(ii),
together
with
[MT],
Lemma
4.2,
(ii),
(v)
[cf.
also
the
discussion
surrounding
[MT],
Lemma
4.2],
that,
relative
to
the
natural
isomor-
∼
phism
I
s
i
→
Z
l
(1),
the
image
of
φ
i
∈
Hom(Π
B
,
Hom
Z
l
(Π
ab
C
,
I
s
i
))
via
the
isomorphisms
∼
ab
Hom(Π
B
,
Hom
Z
l
(Π
ab
C
,
I
s
i
))
→
Hom(Π
B
,
Hom
Z
l
(Π
C
,
Z
l
(1)))
∼
∼
←
H
2
(Π
B
×
Π
C
,
Z
l
(1))
→
H
2
(B
×
k
C,
Z
l
(1))
—
where
the
first
(respectively,
second)
isomorphism
is
the
isomor-
∼
phism
induced
by
the
above
isomorphism
I
s
i
→
Z
l
(1)
(respectively,
the
pr
1
Hochschild-Serre
spectral
sequence
relative
to
the
surjection
Π
B
×Π
C
Π
B
)
—
is
the
first
Chern
class
of
the
invertible
sheaf
associated
to
the
divisor
determined
by
the
scheme-theoretic
image
of
s
i
:
B
i
→
B
×
k
C.
Thus,
since
the
section
s
i
extends
uniquely
to
a
section
s
i
:
B
→
B
×
k
C,
whose
scheme-theoretic
image
we
denote
by
Im(s
i
),
it
follows
that
the
homomorphism
φ
i
∈
Hom(Π
B
,
Hom
Z
l
(Π
ab
C
,
I
s
i
))
coincides
with
the
im-
age
of
the
first
Chern
class
of
the
invertible
sheaf
on
B
×
k
C
associated
to
the
divisor
Im(s
i
)
via
the
composite
∼
H
2
(B
×
k
C,
Z
l
(1))
←
H
2
(B
×
k
C,
I
s
i
)
→
H
2
(B
×
k
C,
I
s
i
)
∼
∼
←
H
2
(Π
B
×
Π
C
,
I
s
i
)
→
Hom
Π
B
,
Hom
Z
l
(Π
C
,
I
s
i
)
—
where
the
first
arrow
is
the
isomorphism
induced
by
the
above
isomor-
∼
phism
I
s
i
→
Z
l
(1),
and
the
second
arrow
is
the
homomorphism
induced
by
the
natural
open
immersion
B
×
k
C
→
B
×
k
C.
In
particular,
as-
sertion
(iii)
follows
immediately
from
[Mln],
Chapter
VI,
Lemma
12.2
[cf.
also
the
argument
used
in
the
proof
of
[MT],
Lemma
4.4].
Finally,
we
verify
assertion
(iv).
To
this
end,
we
recall
that
by
Lemma
1.1,
(iii),
)
factors
through
the
homomor-
the
homomorphism
Π
B
→
Aut
Z
l
(Π
ab
Z/B
n
phism
Π
B
→
Hom
Z
l
Π
ab
of
assertion
(i).
Thus,
assertion
C
,
i=1
J
s
i
(iv)
follows
immediately
from
assertion
(iii).
This
completes
the
proof
of
assertion
(iv).
Q.E.D.
Definition
1.4.
For
∈
{◦,
•},
let
Σ
be
a
set
of
prime
numbers
which
is
either
of
cardinality
one
or
equal
to
the
set
of
all
prime
numbers;
(g
,
r
)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
X
a
hyperbolic
curve
of
type
(g
,
r
)
over
an
algebraically
closed
field
of
Combinatorial
anabelian
topics
I
21
characteristic
∈
Σ
;
d
a
positive
integer;
X
d
the
d
-th
configuration
the
pro-Σ
configura-
space
of
X
[cf.
[MT],
Definition
2.1,
(i)];
Π
d
tion
space
group
[cf.
[MT],
Definition
2.3,
(i)]
obtained
by
forming
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
π
1
(X
d
)
of
X
d
.
∼
(i)
We
shall
say
that
an
isomorphism
of
profinite
groups
α
:
Π
◦
d
◦
→
Π
•
d
•
is
PF-admissible
[i.e.,
“permutation-fiber-admissible”]
if
α
induces
a
bijection
between
the
set
of
fiber
subgroups
[cf.
[MT],
Definition
2.3,
(iii)]
of
Π
◦
d
◦
and
the
set
of
fiber
subgroups
of
Π
•
d
•
.
∼
We
shall
say
that
an
outer
isomorphism
Π
◦
d
◦
→
Π
•
d
•
is
PF-
admissible
if
it
is
determined
by
a
PF-admissible
isomorphism.
(ii)
We
shall
say
that
an
isomorphism
of
profinite
groups
α
:
Π
◦
d
◦
→
Π
•
d
•
is
PC-admissible
[i.e.,
“permutation-cusp-admissible”]
if
the
following
condition
is
satisfied:
Let
∼
{1}
=
K
d
◦
⊆
K
d
◦
−1
⊆
·
·
·
⊆
K
m
⊆
·
·
·
⊆
K
2
⊆
K
1
⊆
K
0
=
Π
◦
d
◦
be
the
standard
fiber
filtration
of
Π
◦
d
◦
[cf.
[CmbCsp],
Definition
1.1,
(i)];
then
for
any
integer
1
≤
a
≤
d
◦
,
the
image
α(K
a
)
⊆
Π
•
d
•
is
a
fiber
subgroup
of
Π
•
d
•
of
length
d
◦
−
a
[cf.
[MT],
Defi-
∼
nition
2.3,
(iii)],
and,
moreover,
the
isomorphism
K
a−1
/K
a
→
α(K
a−1
)/α(K
a
)
determined
by
α
induces
a
bijection
between
the
set
of
cuspidal
inertia
subgroups
of
K
a−1
/K
a
and
the
set
of
cuspidal
inertia
subgroups
of
α(K
a−1
)/α(K
a
).
[Note
that
it
follows
immediately
from
the
various
definitions
involved
that
the
profinite
group
K
a−1
/K
a
(respectively,
α(K
a−1
)/α(K
a
))
is
equipped
with
a
natural
structure
of
pro-Σ
◦
(respectively,
pro-Σ
•
)
surface
group
[cf.
[MT],
Definition
1.2].]
We
shall
say
∼
that
an
outer
isomorphism
Π
◦
d
◦
→
Π
•
d
•
is
PC-admissible
if
it
is
determined
by
a
PC-admissible
isomorphism.
(iii)
∼
We
shall
say
that
an
isomorphism
of
profinite
groups
α
:
Π
◦
d
◦
→
Π
•
d
•
is
PFC-admissible
[i.e.,
“permutation-fiber-cusp-admissi-
ble”]
if
α
is
PF-admissible
and
PC-admissible.
We
shall
say
∼
that
an
outer
isomorphism
Π
◦
d
◦
→
Π
•
d
•
is
PFC-admissible
if
it
is
determined
by
a
PFC-admissible
isomorphism.
22
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(iv)
∼
We
shall
say
that
an
isomorphism
of
profinite
groups
α
:
Π
◦
d
◦
→
Π
•
d
•
is
PF-cuspidalizable
if
there
exists
a
commutative
diagram
∼
Π
◦
d
◦
+1
−−−−→
Π
•
d
•
+1
⏐
⏐
⏐
⏐
Π
◦
d
◦
∼
−−−−→
α
Π
•
d
•
—
where
the
upper
horizontal
arrow
is
a
PF-admissible
iso-
morphism,
and
the
left-hand
(respectively,
right-hand)
vertical
arrow
is
the
surjection
obtained
by
forming
the
quotient
by
a
fiber
subgroup
of
length
1
[cf.
[MT],
Definition
2.3,
(iii)]
of
Π
◦
d
◦
+1
(respectively,
Π
•
d
•
+1
).
We
shall
say
that
an
outer
iso-
∼
morphism
Π
◦
d
◦
→
Π
•
d
•
is
PF-cuspidalizable
if
it
is
determined
by
a
PF-cuspidalizable
isomorphism.
Remark
1.4.1.
It
follows
immediately
from
the
various
definitions
involved
that,
in
the
notation
of
Definition
1.4,
an
automorphism
α
of
Π
◦
d
◦
is
PF-admissible
(respectively,
PC-admissible;
PFC-admissible)
if
and
only
if
there
exists
an
automorphism
σ
of
Π
◦
d
◦
that
lifts
the
outo-
morphism
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
of
Π
◦
d
◦
naturally
determined
by
a
permutation
of
the
d
◦
factors
of
the
config-
uration
space
involved
such
that
the
composite
α
◦
σ
is
F-admissible
(respectively,
C-admissible;
FC-admissible)
[cf.
[CmbCsp],
Definition
1.1,
(ii)].
In
particular,
a(n)
F-admissible
(respectively,
C-admissible;
FC-admissible)
automorphism
of
Π
◦
d
◦
is
PF-admissible
(respectively,
PC-
admissible;
PFC-admissible):
F-admissible
⇐=
FC-admissible
=⇒
C-admissible
⇓
⇓
⇓
PF-admissible
⇐=
PFC-admissible
=⇒
PC-admissible
.
Proposition
1.5
(Properties
of
PF-admissible
isomor-
∼
phisms).
In
the
notation
of
Definition
1.4,
let
α
:
Π
◦
d
◦
→
Π
•
d
•
be
an
isomorphism.
Then
the
following
hold:
(i)
Σ
◦
=
Σ
•
.
Combinatorial
anabelian
topics
I
23
(ii)
Suppose
that
the
isomorphism
α
is
PF-admissible.
Let
1
≤
n
≤
d
◦
be
an
integer
and
H
⊆
Π
◦
d
◦
a
fiber
subgroup
of
length
n
of
Π
◦
d
◦
.
Then
the
subgroup
α(H)
⊆
Π
•
d
•
is
a
fiber
subgroup
of
length
n
of
Π
•
d
•
.
In
particular,
it
holds
that
d
◦
=
d
•
.
(iii)
Write
Ξ
◦
⊆
Π
◦
d
◦
(respectively,
Ξ
•
⊆
Π
•
d
•
)
for
the
normal
closed
subgroup
of
Π
◦
d
◦
(respectively,
Π
•
d
•
)
obtained
by
taking
the
in-
tersection
of
the
various
fiber
subgroups
of
length
d
◦
−
1
(re-
spectively,
d
•
−1).
Then
the
isomorphism
α
is
PF-admissible
∼
if
and
only
if
α
induces
an
isomorphism
Ξ
◦
→
Ξ
•
.
Proof.
Assertion
(i)
follows
immediately
from
the
[easily
verified]
fact
that
Σ
may
be
characterized
as
the
smallest
set
of
primes
Σ
∗
for
is
pro-Σ
∗
.
Assertion
(ii)
follows
immediately
from
the
various
which
Π
d
definitions
involved.
Finally,
we
verify
assertion
(iii).
The
necessity
of
the
condition
follows
immediately
from
assertion
(ii).
The
sufficiency
of
the
condition
follows
immediately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
[CmbCsp],
Proposition
1.2,
(i).
This
completes
the
proof
of
assertion
(iii).
Q.E.D.
Lemma
1.6
(C-admissibility
of
certain
isomorphisms).
In
the
∼
∼
∼
notation
of
Definition
1.4,
let
α
2
:
Π
◦
2
→
Π
•
2
,
α
1
1
:
Π
◦
1
→
Π
•
1
,
α
1
2
:
Π
◦
1
→
•
Π
1
be
isomorphisms
of
profinite
groups
which,
for
i
=
1,
2,
fit
into
a
commutative
diagram
α
2
→
Π
•
2
Π
◦
2
−−−−
⏐
⏐
⏐
⏐
pr
•
pr
◦
{i}
{i}
α
i
1
Π
◦
1
−−−−
→
Π
•
1
—
where
the
vertical
arrow
“pr
{i}
”
is
the
surjection
induced
by
the
pro-
jection
“X
2
→
X
1
”
obtained
by
projecting
to
the
i-th
factor.
Then
the
isomorphism
α
1
1
is
C-admissible.
In
particular,
(g
◦
,
r
◦
)
=
(g
•
,
r
•
).
def
Proof.
Write
Σ
=
Σ
◦
=
Σ
•
[cf.
Proposition
1.5,
(i)].
Now
it
fol-
lows
from
the
well-known
structure
of
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
a
smooth
curve
over
an
algebraically
closed
field
of
characteristic
∈
Σ
that
Π
1
is
a
free
pro-Σ
group
if
and
only
if
r
=
0
[cf.
[CmbGC],
Remark
1.1.3].
Thus,
if
r
◦
=
r
•
=
0,
then
it
is
immediate
that
α
1
1
is
C-admissible;
moreover,
it
follows,
by
considering
the
rank
of
the
abelianization
of
Π
1
[cf.
[CmbGC],
Remark
1.1.3],
that
g
◦
=
g
•
.
In
particular,
to
verify
Lemma
1.6,
we
may
assume
without
loss
24
Yuichiro
Hoshi
and
Shinichi
Mochizuki
of
generality
that
r
◦
,
r
•
=
0.
Then
it
follows
from
[CmbGC],
Theorem
1.6,
(i),
that,
to
verify
Lemma
1.6,
it
suffices
to
show
that
α
1
1
is
numer-
ically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(ii)],
i.e.,
to
show
that
the
following
assertion
holds:
Let
Π
◦
Y
⊆
Π
◦
1
be
an
open
subgroup
of
Π
◦
1
.
Write
def
Π
•
Y
=
α
1
1
(Π
◦
Y
)
⊆
Π
•
1
,
Y
◦
→
X
◦
(respectively,
Y
•
→
X
•
)
for
the
connected
finite
étale
covering
of
X
◦
(re-
spectively,
X
•
)
corresponding
to
the
open
subgroup
Π
◦
Y
⊆
Π
◦
1
(respectively,
Π
•
Y
⊆
Π
•
1
),
and
(g
Y
◦
,
r
Y
◦
)
(re-
spectively,
(g
Y
•
,
r
Y
•
))
for
the
type
of
Y
◦
(respectively,
Y
•
).
Then
it
holds
that
r
Y
◦
=
r
Y
•
.
On
the
other
hand,
in
the
notation
of
the
above
assertion,
one
verifies
easily
that
for
any
l
∈
Σ
and
∈
{◦,
•},
if
Π
Y
⊆
Π
1
is
an
open
subgroup
of
Π
1
contained
in
Π
Y
,
then
the
natural
inclusion
Π
Y
→
Π
Y
induces
a
surjection
ab
ab
ab
Ker((Π
(Π
)
⊗
Z
Σ
Q
l
Ker((Π
(Π
)
ab
)
⊗
Z
Σ
Q
l
)
Y
)
Y
)
Y
Y
—
where
we
write
(Π
),
(Π
)
for
the
maximal
pro-Σ
quotients
of
the
Y
Y
étale
fundamental
groups
of
the
compactifications
Y
,
Y
of
Y
,
Y
,
re-
spectively.
Thus,
since
any
open
subgroup
of
Π
◦
1
contains
a
character-
istic
open
subgroup
of
Π
◦
1
,
it
follows
immediately
from
the
well-known
ab
Σ
-
(respectively,
(Π
)
ab
)
is
a
free
Z
fact
that
for
∈
{◦,
•},
(Π
Y
)
Y
module
of
rank
2g
Y
+
r
Y
−
1
(respectively,
2g
Y
)
[cf.,
e.g.,
[CmbGC],
Remark
1.1.3]
that
to
verify
the
above
assertion,
it
suffices
to
ver-
ify
that
if
Π
◦
Y
⊆
Π
◦
1
in
the
above
assertion
is
characteristic,
then
the
∼
isomorphism
Π
◦
Y
→
Π
•
Y
determined
by
α
1
1
induces
an
isomorphism
of
Ker((Π
◦
Y
)
ab
(Π
◦
Y
)
ab
)
⊗
Z
Σ
Q
l
with
Ker((Π
•
Y
)
ab
(Π
•
Y
)
ab
)
⊗
Z
Σ
Q
l
for
some
l
∈
Σ.
To
this
end,
for
∈
{◦,
•},
write
Π
Z
⊆
Π
2
for
the
normal
open
subgroup
of
Π
2
obtained
by
forming
the
inverse
image
via
the
surjection
(pr
{1}
,
pr
{2}
)
:
Π
2
Π
1
×
Π
1
of
the
image
of
the
natural
inclusion
Π
→
Y
×
Π
Y
→
Π
1
×
Π
1
;
Z
X
2
for
the
connected
finite
étale
covering
corresponding
to
this
normal
open
subgroup
Π
Z
⊆
Π
2
;
Π
Z/Y
for
the
kernel
of
the
natural
surjection
pr
→
1
X
.
Π
Z
Π
Y
induced
by
the
composite
Z
→
X
2
→
X
×
k
X
Then
the
natural
surjection
Π
Z
Π
Y
determines
a
representation
ab
Π
Y
−→
Aut((Π
Z/Y
)
)
;
Combinatorial
anabelian
topics
I
25
moreover,
the
isomorphisms
α
2
,
α
1
1
,
and
α
1
2
determine
a
commutative
diagram
Π
◦
Y
−−−−→
Aut((Π
◦
Z/Y
)
ab
)
⏐
⏐
⏐
⏐
Π
•
Y
−−−−→
Aut((Π
•
Z/Y
)
ab
)
—
where
the
vertical
arrows
are
isomorphisms.
[Here,
we
note
that
since
Π
•
Y
is
a
characteristic
subgroup
of
Π
•
1
,
and
the
composite
α
1
2
◦
(α
1
1
)
−1
is
an
automorphism
of
Π
•
1
,
it
follows
that
Π
•
Y
=
α
1
2
(Π
◦
Y
),
hence
that
∼
α
2
induces
an
isomorphism
Π
◦
Z
→
Π
•
Z
.]
On
the
other
hand,
it
follows
from
the
definition
of
Z
that
Z
is
isomorphic
to
the
open
subscheme
of
Y
×
k
Y
obtained
by
forming
the
complement
of
the
graphs
of
the
various
elements
of
Aut(Y
/X
).
Thus,
it
follows
from
Lemma
1.3,
(iv)
—
by
replacing
the
various
profinite
groups
involved
by
their
maximal
∼
pro-l
quotients
for
some
l
∈
Σ
—
that
the
isomorphism
Π
◦
Y
→
Π
•
Y
1
◦
ab
◦
ab
determined
by
α
1
induces
an
isomorphism
of
Ker((Π
Y
)
(Π
Y
)
)⊗
Z
Σ
Q
l
with
Ker((Π
•
Y
)
ab
(Π
•
Y
)
ab
)
⊗
Z
Σ
Q
l
for
some
l
∈
Σ.
This
completes
the
proof
of
Lemma
1.6.
Q.E.D.
Lemma
1.7
(PFC-admissibility
of
certain
PF-admissible
iso-
∼
morphisms).
In
the
notation
of
Definition
1.4,
let
α
:
Π
◦
d
◦
→
Π
•
d
•
be
a
PF-admissible
isomorphism.
Then
the
following
condition
implies
that
the
isomorphism
α
is
PFC-admissible:
Let
H
◦
⊆
Π
◦
d
◦
be
a
fiber
subgroup
of
length
1
[cf.
def
[MT],
Definition
2.3,
(iii)].
Write
H
•
=
α(H
◦
)
⊆
Π
•
d
•
for
the
fiber
subgroup
of
length
1
obtained
as
the
image
of
H
◦
via
α
[cf.
Proposition
1.5,
(ii)].
[Thus,
it
follows
immediately
from
the
various
defini-
tions
involved
that
H
◦
(respectively,
H
•
)
is
equipped
with
a
natural
structure
of
pro-Σ
◦
(respectively,
pro-
∼
Σ
•
)
surface
group.]
Then
the
isomorphism
H
◦
→
H
•
induced
by
α
is
C-admissible.
Proof.
Let
∈
{◦,
•}.
Then
one
may
verify
easily
that
the
follow-
ing
fact
holds:
Let
1
≤
a
≤
d
be
an
integer
and
F
⊆
F
⊆
Π
d
fiber
subgroups
of
Π
such
that
F
is
of
length
a,
d
and
F
is
of
length
a
−
1.
Then
there
exists
a
fiber
26
Yuichiro
Hoshi
and
Shinichi
Mochizuki
subgroup
H
⊆
F
⊆
Π
of
Π
of
length
1
such
that
d
d
the
composite
H
→
F
F/F
arises
from
a
natural
open
immersion
of
a
hyperbolic
curve
of
type
(g
,
r
+d
−1)
into
a
hyperbolic
curve
of
type
(g
,
r
+
d
−
a).
[Note
that
it
follows
im-
mediately
from
the
various
definitions
involved
that
H
(respectively,
F/F
)
is
equipped
with
a
natural
structure
of
pro-Σ
surface
group.]
In
particular,
the
composite
is
a
surjection
whose
kernel
is
topologically
normally
generated
by
suitable
cuspidal
inertia
sub-
groups
of
H;
moreover,
any
cuspidal
inertia
subgroup
of
F/F
may
be
obtained
as
the
image
of
a
cuspidal
inertia
subgroup
of
H.
On
the
other
hand,
one
may
verify
easily
that
Lemma
1.7
follows
im-
mediately
from
the
above
fact.
This
completes
the
proof
of
Lemma
1.7.
Q.E.D.
Theorem
1.8
(PFC-admissibility
of
certain
isomorphisms).
For
∈
{◦,
•},
let
Σ
be
a
set
of
prime
numbers
which
is
either
of
cardinality
one
or
equal
to
the
set
of
all
prime
numbers;
(g
,
r
)
a
pair
of
nonnegative
integers
such
that
2g
−2+r
>
0;
X
a
hyperbolic
curve
of
type
(g
,
r
)
over
an
algebraically
closed
field
of
characteristic
the
pro-Σ
configuration
space
group
∈
Σ
;
d
a
positive
integer;
Π
d
[cf.
[MT],
Definition
2.3,
(i)]
obtained
by
forming
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
of
the
d
-th
configuration
space
of
X
;
∼
α
:
Π
◦
d
◦
−→
Π
•
d
•
an
isomorphism
of
[abstract]
groups.
If
∅
,
{(g
◦
,
r
◦
),
(g
•
,
r
•
)}
∩
{(0,
3),
(1,
1)}
=
then
we
suppose
further
that
the
isomorphism
α
is
PF-admissible
[cf.
Definition
1.4,
(i)].
Then
the
following
hold:
(i)
Σ
◦
=
Σ
•
.
(ii)
The
isomorphism
α
is
an
isomorphism
of
profinite
groups.
(iii)
The
isomorphism
α
is
PF-admissible.
In
particular,
d
◦
=
d
•
.
Combinatorial
anabelian
topics
I
(iv)
27
If
α
is
PF-cuspidalizable
[cf.
Definition
1.4,
(iv)],
then
α
is
PFC-admissible
[cf.
Definition
1.4,
(iii)].
In
particular,
(g
◦
,
r
◦
)
=
(g
•
,
r
•
).
Proof.
Assertion
(ii)
follows
from
[NS],
Theorem
1.1.
In
light
of
assertion
(ii),
assertion
(i)
follows
from
Proposition
1.5,
(i).
Assertion
(iii)
follows
from
Proposition
1.5,
(ii);
[MT],
Corollary
6.3,
together
with
the
assumption
appearing
in
the
statement
of
Theorem
1.8.
Assertion
(iv)
follows
immediately
from
Lemmas
1.6,
1.7.
Q.E.D.
Corollary
1.9
(F-admissibility
and
FC-admissibility).
Let
Σ
be
a
set
of
prime
numbers
which
is
either
of
cardinality
one
or
equal
to
the
set
of
all
prime
numbers;
n
a
positive
integer;
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
X
a
hyperbolic
curve
of
type
(g,
r)
over
an
algebraically
closed
field
k
of
characteristic
∈
Σ;
X
n
the
n-th
configuration
space
of
X;
Π
n
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
X
n
;
“Out
FC
(−)”,
“Out
F
(−)”
⊆
“Out(−)”
the
subgroups
of
FC-
and
F-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
outomorphisms
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
of
“(−)”.
Then
the
following
hold:
(i)
Let
α
∈
Out
F
(Π
n+1
).
Then
α
induces
the
same
outomor-
phism
of
Π
n
relative
to
the
various
quotients
Π
n+1
Π
n
by
fiber
subgroups
of
length
1
[cf.
[MT],
Definition
2.3,
(iii)].
In
particular,
we
obtain
a
natural
homomorphism
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
.
(ii)
The
image
of
the
homomorphism
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
of
(i)
is
contained
in
Out
FC
(Π
n
)
⊆
Out
F
(Π
n
)
.
Proof.
First,
we
verify
assertion
(i).
Let
H
1
,
H
2
⊆
Π
n+1
be
two
distinct
fiber
subgroups
of
Π
n+1
of
length
1.
Observe
that
the
normal
closed
subgroup
H
⊆
Π
n+1
of
Π
n+1
topologically
generated
by
H
1
and
H
2
is
a
fiber
subgroup
of
Π
n+1
of
length
2
[cf.
[MT],
Proposition
2.4,
(iv)],
hence
is
equipped
with
a
natural
structure
of
pro-Σ
configuration
space
group,
with
respect
to
which
H
i
⊆
H
may
be
regarded
as
a
fiber
28
Yuichiro
Hoshi
and
Shinichi
Mochizuki
subgroup
of
length
1
[cf.
[MT],
Proposition
2.4,
(ii)].
Moreover,
it
follows
immediately
from
the
scheme-theoretic
definition
of
the
various
config-
uration
space
groups
involved
that
one
has
natural
outer
isomorphisms
∼
∼
Π
n+1
/H
i
→
Π
n
and
H/H
1
→
H/H
2
.
Thus,
since
for
i
∈
{1,
2},
we
have
natural
outer
isomorphisms
∼
∼
out
Π
n
←
Π
n+1
/H
i
→
(H/H
i
)
Π
n+1
/H
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
which
are
com-
patible
with
the
various
natural
outer
isomorphisms
discussed
above,
one
verifies
easily
[cf.
the
argument
given
in
the
first
paragraph
of
the
proof
of
[CmbCsp],
Theorem
4.1]
that
to
complete
the
proof
of
assertion
(i),
by
replacing
Π
n+1
by
H,
it
suffices
to
verify
assertion
(i)
in
the
case
where
n
=
1.
The
rest
of
the
proof
of
assertion
(i)
is
devoted
to
verifying
assertion
(i)
in
the
case
where
n
=
1.
1
,
Let
α
∈
Aut
F
(Π
2
)
be
an
F-admissible
automorphism
of
Π
2
;
α
2
relative
to
the
α
∈
Aut(Π
1
)
the
automorphisms
of
Π
1
induced
by
α
∼
∼
quotients
Π
2
Π
2
/H
1
→
Π
1
,
Π
2
Π
2
/H
2
→
Π
1
,
respectively.
Now
it
is
immediate
that
to
complete
the
proof
of
assertion
(i),
it
suffices
to
α
2
)
−1
∈
Aut(Π
1
)
is
Π
1
-inner.
Therefore,
verify
that
the
difference
α
1
◦
(
it
follows
immediately
from
[JR],
Theorem
B,
that
to
complete
the
proof
of
assertion
(i),
it
suffices
to
verify
that
(∗
1
):
for
any
normal
open
subgroup
N
⊆
Π
1
of
Π
1
,
2
(N
).
it
holds
that
α
1
(N
)
=
α
To
this
end,
let
N
⊆
Π
1
be
a
normal
open
subgroup
of
Π
1
.
Write
∼
def
Π
N
=
Π
2
×
Π
1
N
for
the
fiber
product
of
Π
2
Π
2
/H
1
→
Π
1
and
pr
1
N
→
Π
1
and
F
N
for
the
kernel
of
the
composite
Π
N
=
Π
2
×
Π
1
N
→
∼
Π
2
Π
2
/H
2
→
Π
1
.
Then
the
surjection
Π
N
N
×
Π
1
determined
by
∼
the
natural
surjection
Π
N
Π
N
/F
N
→
Π
1
and
the
second
projection
pr
2
Π
N
=
Π
2
×
Π
1
N
N
fits
into
a
commutative
diagram
of
profinite
groups
1
−−−−→
F
N
−−−−→
⏐
⏐
Π
N
⏐
⏐
−−−−→
Π
1
−−−−→
1
pr
2
→
Π
1
−−−−→
1
1
−−−−→
N
−−−−→
N
×
Π
1
−−−−
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
surjective.
Write
ρ
N
:
Π
1
→
Aut(F
N
ab
)
for
the
natural
action
determined
by
the
upper
horizontal
sequence
and
V
N
⊆
F
N
ab
for
the
kernel
of
the
natural
surjection
F
N
ab
N
ab
induced
by
the
left-hand
vertical
arrow.
Now
we
claim
that
Combinatorial
anabelian
topics
I
29
(∗
2
):
the
action
ρ
N
of
Π
1
on
F
N
ab
preserves
V
N
⊆
F
N
ab
,
and,
moreover,
the
resulting
action
ρ
VN
:
Π
1
→
Aut(V
N
)
factors
as
the
composite
Π
1
Π
1
/N
→
Aut(V
N
)
—
where
the
second
arrow
is
injective.
Indeed,
the
fact
that
the
action
ρ
N
of
Π
1
on
F
N
ab
preserves
V
N
⊆
F
N
ab
follows
immediately
from
the
definition
of
ρ
N
[cf.
also
the
above
com-
mutative
diagram].
Next,
let
us
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
if
we
write
f
:
Y
→
X
for
the
con-
nected
finite
étale
Galois
covering
of
X
corresponding
to
N
⊆
Π
1
,
then
the
right-hand
square
of
the
above
diagram
arises
from
a
commutative
diagram
of
schemes
pr
2
→
X
(Y
×
k
X)
\
Γ
f
−−−−
⏐
⏐
Y
×
k
X
pr
2
−−−−
→
X
—
where
we
write
Γ
f
⊆
Y
×
k
X
for
the
graph
of
f
,
and
the
left-hand
vertical
arrow
is
the
natural
open
immersion.
Thus,
it
follows
imme-
diately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
Lemma
1.1,
(i)
[cf.
also
[Hsh],
Proposition
1.4,
(i)],
that
F
N
,
N
are
naturally
isomorphic
to
the
maximal
pro-Σ
quotients
of
the
étale
fun-
damental
groups
of
geometric
fibers
of
the
families
of
hyperbolic
curves
pr
Y
×
k
X
\
Γ
f
,
Y
×
k
X
→
2
X
over
X,
respectively.
Therefore,
by
the
well-known
structure
of
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
a
smooth
curve
over
an
algebraically
closed
field
of
character-
istic
∈
Σ,
we
conclude
—
by
considering
the
natural
action
of
Π
1
on
the
pr
set
of
cusps
of
the
family
of
hyperbolic
curves
Y
×
k
X
\
Γ
f
→
2
X
—
that
the
resulting
action
ρ
VN
:
Π
1
→
Aut(V
N
)
factors
as
the
composite
Π
1
Π
1
/N
→
Aut(V
N
),
and
that
if
X
is
affine
(respectively,
proper),
then
for
any
l
∈
Σ,
the
resulting
representation
Π
1
/N
→
Aut(V
N
⊗
Z
Σ
Q
l
)
is
isomorphic
to
the
regular
representation
of
Π
1
/N
over
Q
l
(respec-
tively,
the
quotient
of
the
regular
representation
of
Π
1
/N
over
Q
l
by
the
trivial
subrepresentation
[of
di-
mension
1]).
In
particular,
as
is
well-known,
the
homomorphism
Π
1
/N
→
Aut(V
N
⊗
Z
Σ
Q
l
),
hence
also
the
homomorphism
Π
1
/N
→
Aut(V
N
),
is
injective.
This
completes
the
proof
of
the
claim
(∗
2
).
30
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Next,
let
us
observe
that
since
α
is
F-admissible,
it
follows
imme-
induces
a
diately
from
the
definition
of
“ρ
VN
”
that
the
automorphism
α
commutative
diagram
ρ
V
Π
1
−−−
N
−→
⏐
⏐
α
2
Aut(V
N
)
⏐
⏐
ρ
V
α
1
(N
)
Π
1
−−−−→
Aut(V
α
1
(N
)
)
—
where
the
vertical
arrows
are
isomorphisms
that
are
induced
by
α
.
Thus,
by
considering
the
kernels
of
ρ
VN
,
ρ
V
α
1
(N
)
,
one
concludes
from
the
claim
(∗
2
)
that
α
1
(N
)
=
α
2
(N
).
This
completes
the
proof
of
(∗
1
),
hence
also
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
Theorem
1.8,
(iv)
[cf.
also
Remark
1.4.1].
This
completes
the
proof
of
Corollary
1.9.
Q.E.D.
Remark
1.9.1.
The
discrete
versions
of
Theorem
1.8,
Corollary
1.9
will
be
discussed
in
a
sequel
to
the
present
paper.
§2.
Various
operations
on
semi-graphs
of
anabelioids
of
PSC-
type
In
the
present
§,
we
study
various
operations
on
semi-graphs
of
anabelioids
of
PSC-type.
These
operations
include
the
following:
(Op1)
the
operation
of
restriction
to
a
sub-semi-graph
[satisfying
cer-
tain
conditions]
of
the
underlying
semi-graph
[cf.
Definition
2.2,
(ii);
Fig.
2
below],
(Op2)
the
operation
of
partial
compactification
[cf.
Definition
2.4,
(ii);
Fig.
3
below],
(Op3)
the
operation
of
resolution
of
a
given
set
[satisfying
certain
conditions]
of
nodes
[cf.
Definition
2.5,
(ii);
Fig.
4
below],
and
(Op4)
the
operation
of
generization
[cf.
Definition
2.8;
Fig.
5
below].
A
basic
reference
for
the
theory
of
semi-graphs
of
anabelioids
of
PSC-
type
is
[CmbGC].
We
shall
use
the
terms
“semi-graph
of
anabelioids
of
PSC-type”,
“PSC-fundamental
group
of
a
semi-graph
of
anabelioids
of
Combinatorial
anabelian
topics
I
31
PSC-type”,
“finite
étale
covering
of
semi-graphs
of
anabelioids
of
PSC-
type”,
“vertex”,
“edge”,
“cusp”,
“node”,
“verticial
subgroup”,
“edge-like
subgroup”,
“nodal
subgroup”,
“cuspidal
subgroup”,
and
“sturdy”
as
they
are
defined
in
[CmbGC],
Definition
1.1.
Also,
we
shall
apply
the
vari-
ous
notational
conventions
established
in
[NodNon],
Definition
1.1,
and
refer
to
the
“PSC-fundamental
group
of
a
semi-graph
of
anabelioids
of
PSC-type”
simply
as
the
“fundamental
group”
[of
the
semi-graph
of
an-
abelioids
of
PSC-type].
That
is
to
say,
we
shall
refer
to
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[as
a
semi-graph
of
anabelioids!]
as
the
“fundamental
group
of
the
semi-graph
of
anabelioids
of
PSC-type”.
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Write
G
for
the
underlying
semi-graph
of
G,
Π
G
for
the
[pro-Σ]
fundamental
group
of
G,
and
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Then
since
the
fundamental
group
Π
G
of
G
is
topologically
finitely
generated,
the
profinite
topology
of
Π
G
induces
[profinite]
topologies
on
Aut(Π
G
)
and
Out(Π
G
)
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
If,
moreover,
we
write
Aut(G)
for
the
automorphism
group
of
G,
then
by
the
discussion
preceding
[CmbGC],
Lemma
2.1,
the
natural
homomorphism
Aut(G)
−→
Out(Π
G
)
is
an
injection
with
closed
image.
[Here,
we
recall
that
an
automorphism
of
a
semi-graph
of
anabelioids
consists
of
an
automorphism
of
the
un-
derlying
semi-graph,
together
with
a
compatible
system
of
isomorphisms
between
the
various
anabelioids
at
each
of
the
vertices
and
edges
of
the
underlying
semi-graph
which
are
compatible
with
the
various
morphisms
of
anabelioids
associated
to
the
branches
of
the
underlying
semi-graph
—
cf.
[SemiAn],
Definition
2.1;
[SemiAn],
Remark
2.4.2.]
Thus,
by
equipping
Aut(G)
with
the
topology
induced
via
this
homomorphism
by
the
topology
of
Out(Π
G
),
we
may
regard
Aut(G)
as
being
equipped
with
the
structure
of
a
profinite
group.
Definition
2.1.
(i)
For
z
∈
VCN(G)
such
that
z
∈
Vert(G)
(respectively,
z
∈
Edge(G)),
we
shall
say
that
a
closed
subgroup
of
Π
G
is
a
VCN-
subgroup
of
Π
G
associated
to
z
∈
VCN(G)
if
the
closed
sub-
group
is
a
verticial
(respectively,
an
edge-like)
subgroup
of
32
Yuichiro
Hoshi
and
Shinichi
Mochizuki
such
that
Π
G
associated
to
z
∈
VCN(G).
For
z
∈
VCN(
G)
(respectively,
z
∈
Edge(
G)),
we
shall
say
that
a
z
∈
Vert(
G)
closed
subgroup
of
Π
G
is
the
VCN-subgroup
of
Π
G
associated
if
the
closed
subgroup
is
the
verticial
(respec-
to
z
∈
VCN(
G)
[cf.
tively,
edge-like)
subgroup
of
Π
G
associated
to
z
∈
VCN(
G)
[NodNon],
Definition
1.1,
(vi)].
(ii)
For
z
∈
VCN(G),
we
shall
write
G
z
for
the
anabelioid
corresponding
to
z
∈
VCN(G).
(iii)
For
v
∈
Vert(G),
we
shall
write
G|
v
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
defined
as
follows
[cf.
Fig.
1
below]:
We
take
Vert(G|
v
)
to
consist
of
the
single
element
“v”,
Cusp(G|
v
)
to
be
the
set
of
branches
of
G
which
abut
to
v,
and
Node(G|
v
)
to
be
the
empty
set.
We
take
the
anabelioid
of
G|
v
corresponding
to
the
unique
vertex
“v”
to
be
G
v
[cf.
(ii)].
For
each
edge
e
∈
E(v)
of
G
and
each
branch
b
of
e
that
abuts
to
the
vertex
v,
we
take
the
anabelioid
of
G|
v
corresponding
to
the
branch
b
to
be
a
copy
of
the
anabelioid
G
e
[cf.
(ii)].
For
each
edge
e
∈
E(v)
of
G
and
each
branch
b
of
e
that
abuts,
relative
to
G,
to
the
vertex
v,
we
take
the
morphism
of
anabelioids
(G|
v
)
e
b
→
(G|
v
)
v
of
G|
v
—
where
we
write
e
b
for
the
cusp
of
G|
v
corresponding
to
b
—
to
be
the
morphism
of
anabelioids
G
e
→
G
v
associated,
relative
to
G,
to
the
branch
b.
Thus,
one
has
a
natural
morphism
G|
v
−→
G
of
semi-graphs
of
anabelioids.
Remark
2.1.1.
Let
v
∈
Vert(G)
be
a
vertex
of
G
and
Π
v
⊆
Π
G
a
verticial
subgroup
of
Π
G
associated
to
v
∈
Vert(G).
Then
it
follows
immediately
from
the
various
definitions
involved
that
the
fundamental
group
of
G|
v
is
naturally
isomorphic
to
Π
v
,
and
that
we
have
a
natural
identification
Aut(G
v
)
≃
Out(Π
v
)
Combinatorial
anabelian
topics
I
33
and
a
natural
injection
Aut(G|
v
)
→
Aut(G
v
)
.
•
G
◦
◦
v
◦
⇒
G|
v
•
◦
•
◦
Figure
1:
G|
v
Definition
2.2
(cf.
the
operation
(Op1)
discussed
at
the
beginning
the
present
§2).
(i)
Let
K
be
a
[not
necessarily
finite]
semi-graph
and
H
a
sub-semi-
graph
of
K
[cf.
[SemiAn],
the
discussion
following
the
figure
entitled
“A
Typical
Semi-graph”].
Then
we
shall
say
that
H
is
of
PSC-type
if
the
following
three
conditions
are
satisfied:
(1)
H
is
finite
[i.e.,
the
set
consisting
of
vertices
and
edges
of
H
is
finite]
and
connected.
(2)
H
has
at
least
one
vertex.
(3)
If
v
is
a
vertex
of
H,
and
e
is
an
edge
of
K
that
abuts
to
v,
then
e
is
an
edge
of
H.
[Thus,
if
e
abuts
both
to
a
vertex
lying
in
H
and
to
a
vertex
not
lying
in
H,
then
the
resulting
edge
of
H
is
a
“cusp”,
i.e.,
an
open
edge.]
Thus,
a
sub-semi-graph
of
PSC-type
H
is
completely
deter-
mined
by
the
set
of
vertices
that
lie
in
H.
(ii)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
(i)]
of
G.
Then
one
may
verify
easily
that
the
semi-graph
of
anabelioids
obtained
by
restricting
G
to
H
[cf.
the
discussion
preceding
[SemiAn],
Definition
2.2]
is
of
pro-Σ
PSC-type.
Here,
we
recall
that
the
semi-graph
of
anabelioids
obtained
by
restricting
G
to
H
is
the
semi-graph
of
anabelioids
such
that
the
underlying
semi-graph
is
H;
for
each
vertex
v
(respectively,
edge
e)
of
H,
the
anabe-
lioid
corresponding
to
v
(respectively,
e)
is
G
v
(respectively,
34
Yuichiro
Hoshi
and
Shinichi
Mochizuki
G
e
)
[cf.
Definition
2.1,
(ii)];
for
each
branch
b
of
an
edge
e
of
H
that
abuts
to
a
vertex
v
of
H,
the
morphism
associated
to
b
is
the
morphism
G
e
→
G
v
associated
to
the
branch
of
G
corresponding
to
b.
We
shall
write
G|
H
for
this
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
and
refer
to
G|
H
as
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
ob-
tained
by
restricting
G
to
H
[cf.
Fig.
2
below].
Thus,
one
has
a
natural
morphism
G|
H
−→
G
of
semi-graphs
of
anabelioids.
v
×
×
G
H:
the
sub-semi-graph
of
PSC-type
whose
set
of
vertices
=
{v}
⇓
×
×
G|
H
Figure
2:
Restriction
Combinatorial
anabelian
topics
I
35
Definition
2.3.
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0.
(i)
We
shall
say
that
G
is
of
type
(g,
r)
if
G
arises
from
a
stable
log
curve
of
type
(g,
r)
over
an
algebraically
closed
field
of
charac-
teristic
∈
Σ,
i.e.,
Cusp(G)
is
of
cardinality
r,
and,
moreover,
rank
Z
Σ
(Π
ab
G
)
=
2g
+
Cusp(G)
−
c
G
—
where
def
c
G
=
0
if
Cusp(G)
=
∅,
1
if
Cusp(G)
=
∅.
[Here,
we
recall
that
it
follows
from
the
discussion
of
[CmbGC],
Σ
Remark
1.1.3,
that
Π
ab
G
is
a
free
Z
-module
of
finite
rank.]
(ii)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)].
Then
we
shall
say
that
H
is
of
type
(g,
r)
if
the
semi-graph
of
anabelioids
G|
H
,
which
is
of
pro-Σ
PSC-type
[cf.
Definition
2.2,
(ii)],
is
of
type
(g,
r)
[cf.
(i)].
(iii)
Let
v
∈
Vert(G)
be
a
vertex.
Then
we
shall
say
that
v
is
of
type
(g,
r)
if
the
semi-graph
of
anabelioids
G|
v
,
which
is
of
pro-Σ
PSC-type
[cf.
Definition
2.1,
(iii)],
is
of
type
(g,
r)
[cf.
(i)].
(iv)
We
shall
say
that
G
is
totally
degenerate
if
each
vertex
of
G
is
of
type
(0,
3)
[cf.
(iii)].
(v)
One
may
verify
easily
that
there
exists
a
unique,
up
to
isomor-
phism,
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
that
is
of
type
(g,
r)
[cf.
(i)]
and
has
no
node.
We
shall
write
model
G
g,r
for
this
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Remark
2.3.1.
It
follows
immediately
from
the
various
definitions
involved
that
there
exists
a
unique
pair
(g,
r)
of
nonnegative
integers
such
that
G
is
of
type
(g,
r)
[cf.
Definition
2.3,
(i)].
Definition
2.4
(cf.
the
operation
(Op2)
discussed
at
the
beginning
the
present
§2).
36
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(i)
We
shall
say
that
a
subset
S
⊆
Cusp(G)
of
Cusp(G)
is
omit-
table
if
the
following
condition
is
satisfied:
For
each
vertex
v
∈
Vert(G)
of
G,
if
v
is
of
type
(g,
r)
[cf.
Definition
2.3,
(iii);
Remark
2.3.1],
then
it
holds
that
2g
−
2
+
r
−
(E(v)
∩
S)
>
0.
(ii)
Let
S
⊆
Cusp(G)
be
a
subset
of
Cusp(G)
which
is
omittable
[cf.
(i)].
Then
by
eliminating
the
cusps
[i.e.,
the
open
edges]
contained
in
S,
and,
for
each
vertex
v
of
G,
replacing
the
an-
abelioid
G
v
corresponding
to
v
by
the
anabelioid
of
finite
étale
coverings
of
G
v
that
restrict
to
a
trivial
covering
over
the
cusps
contained
in
S
that
abut
to
v,
we
obtain
a
semi-graph
of
an-
abelioids
G
•S
of
pro-Σ
PSC-type.
We
shall
refer
to
G
•S
as
the
partial
compact-
ification
of
G
with
respect
to
S
[cf.
Fig.
3
below].
Thus,
for
each
v
∈
Vert(G)
=
Vert(G
•S
),
the
pro-Σ
fundamental
group
of
the
anabelioid
(G
•S
)
v
corresponding
to
v
∈
Vert(G)
=
Vert(G
•S
)
may
be
naturally
identified,
up
to
inner
automorphism,
with
the
quotient
of
a
verticial
subgroup
Π
v
⊆
Π
G
of
Π
G
associated
to
v
∈
Vert(G)
=
Vert(G
•S
)
by
the
subgroup
of
Π
v
topolog-
ically
normally
generated
by
the
Π
e
⊆
Π
v
for
e
∈
E(v)
∩
S.
If,
moreover,
we
write
Π
G
•S
for
the
[pro-Σ]
fundamental
group
of
G
•S
and
N
S
⊆
Π
G
for
the
normal
closed
subgroup
of
Π
G
topologically
normally
generated
by
the
cuspidal
subgroups
of
Π
G
associated
to
elements
of
S,
then
we
have
a
natural
outer
isomorphism
∼
Π
G
/N
S
−→
Π
G
•S
.
Remark
2.4.1.
(i)
Let
S
1
⊆
S
2
⊆
Cusp(G)
be
subsets
of
Cusp(G).
Then
it
fol-
lows
immediately
from
the
various
definitions
involved
that
the
omittability
of
S
2
[cf.
Definition
2.4,
(i)]
implies
the
omittability
of
S
1
.
(ii)
If
G
is
sturdy,
then
it
follows
from
the
various
definitions
in-
volved
that
Cusp(G),
hence
also
any
subset
of
Cusp(G)
[cf.
(i)],
is
omittable.
Moreover,
the
partial
compactification
of
G
with
Combinatorial
anabelian
topics
I
37
respect
to
Cusp(G)
coincides
with
the
compactification
of
G
[cf.
[CmbGC],
Remark
1.1.6;
[NodNon],
Definition
1.11].
c
1
c
2
×
×
×
G
⇓
×
G
•{c
1
,c
2
}
Figure
3:
Partial
compactification
Definition
2.5
(cf.
the
operation
(Op3)
discussed
at
the
beginning
the
present
§2).
Let
S
⊆
Node(G)
be
a
subset
of
Node(G).
(i)
We
shall
say
that
S
is
of
separating
type
if
the
semi-graph
obtained
by
removing
the
closed
edges
corresponding
to
the
elements
of
S
from
G
is
disconnected.
Moreover,
for
each
node
e
∈
Node(G),
we
shall
say
that
e
is
of
separating
type
if
{e}
⊆
Node(G)
is
of
separating
type.
(ii)
Suppose
that
S
is
not
of
separating
type
[cf.
(i)].
Then
one
may
define
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
as
follows:
We
take
the
underlying
semi-graph
G
S
to
be
the
semi-graph
obtained
by
replacing
each
node
e
of
G
contained
in
S
such
that
V(e)
=
{v
1
,
v
2
}
⊆
Vert(G)
—
where
v
1
,
v
2
are
not
necessarily
distinct
—
by
two
cusps
that
abut
to
v
1
,
v
2
∈
Vert(G),
respectively.
We
take
the
anabelioid
correspond-
ing
to
a
vertex
v
(respectively,
node
e)
of
G
S
to
be
G
v
(respec-
tively,
G
e
).
[Note
that
the
set
of
vertices
(respectively,
nodes)
38
Yuichiro
Hoshi
and
Shinichi
Mochizuki
of
G
S
may
be
naturally
identified
with
Vert(G)
(respectively,
Node(G)
\
S).]
We
take
the
anabelioid
corresponding
to
a
cusp
of
G
S
arising
from
a
cusp
e
of
G
to
be
G
e
.
We
take
the
an-
abelioid
corresponding
to
a
cusp
of
G
S
arising
from
a
node
e
of
G
to
be
G
e
.
For
each
branch
b
of
G
S
that
abuts
to
a
vertex
v
of
a
node
e
(respectively,
of
a
cusp
e
that
does
not
arise
from
a
node
of
G),
we
take
the
morphism
associated
to
b
to
be
the
morphism
G
e
→
G
v
associated
to
the
branch
of
G
corresponding
to
b.
For
each
branch
b
of
G
S
that
abuts
to
a
vertex
v
of
a
cusp
of
G
S
that
arises
from
a
node
e
of
G,
we
take
the
morphism
associated
to
b
to
be
the
morphism
G
e
→
G
v
associated
to
the
branch
of
G
corresponding
to
b.
We
shall
denote
the
resulting
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
by
G
S
and
refer
to
G
S
as
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
obtained
from
G
by
resolving
S
[cf.
Fig.
4
below].
Thus,
one
has
a
natural
morphism
G
S
−→
G
of
semi-graphs
of
anabelioids.
Remark
2.5.1.
(i)
Let
S
1
⊆
S
2
⊆
Node(G)
be
subsets
of
Node(G).
Then
it
follows
immediately
from
the
various
definitions
involved
that
if
S
2
is
not
of
separating
type
[cf.
Definition
2.5,
(i)],
then
S
1
is
not
of
separating
type.
(ii)
Let
v
∈
Vert(G)
be
a
vertex
of
G.
Then
one
may
verify
eas-
ily
that
there
exists
a
unique
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
G
v
of
G
such
that
the
set
of
vertices
of
G
v
is
equal
to
{v}.
Moreover,
one
may
also
verify
easily
that
Node(G|
G
v
)
[cf.
Definition
2.2,
(ii)]
is
not
of
separating
type
[cf.
Definition
2.5,
(i)],
relative
to
G|
G
v
,
and
that
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
(G|
G
v
)
Node(G|
G
v
)
[cf.
Definition
2.5,
(ii)]
is
naturally
isomorphic
to
G|
v
[cf.
Def-
inition
2.1,
(iii)].
Combinatorial
anabelian
topics
I
39
×
×
G
e
⇓
×
×
×
×
G
{e}
Figure
4:
Resolution
40
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Definition
2.6.
(i)
Let
S
⊆
VCN(G)
be
a
subset
of
VCN(G).
Then
we
shall
denote
by
Aut
S
(G)
⊆
Aut(G)
the
[closed]
subgroup
of
Aut(G)
consisting
of
automorphisms
α
of
G
such
that
the
automorphism
of
the
underlying
semi-graph
G
of
G
induced
by
α
preserves
S
and
by
Aut
|S|
(G)
⊆
Aut
S
(G)
the
[closed]
subgroup
of
Aut(G)
consisting
of
automorphisms
α
of
G
such
that
the
automorphism
of
the
underlying
semi-
graph
G
of
G
induced
by
α
preserves
and
induces
the
identity
automorphism
of
S.
Moreover,
we
shall
write
Aut
|grph|
(G)
=
Aut
|VCN(G)|
(G)
.
def
(ii)
Let
H
⊆
Π
G
be
a
closed
subgroup
of
Π
G
.
Then
we
shall
denote
by
Out
H
(Π
G
)
⊆
Out(Π
G
)
the
[closed]
subgroup
of
Out(Π
G
)
consisting
of
outomorphisms
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
of
Π
G
which
preserve
the
Π
G
-conjugacy
class
of
H
⊆
Π
G
.
Moreover,
we
shall
denote
by
def
Aut
H
(G)
=
Aut(G)
∩
Out
H
(Π
G
)
.
(iii)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G.
Then
VCN(G|
H
)
[cf.
Definition
2.2,
(ii)]
may
be
regarded
as
a
subset
of
VCN(G).
We
shall
write
Aut
|H|
(G)
def
=
Aut
|VCN(G|
H
)|
(G)
⊆
Aut
H
(G)
def
=
=
Aut
VCN(G|
H
)
(G)
Aut
Vert(G|
H
)
(G)
.
Proposition
2.7
(Subgroups
determined
by
sets
of
compo-
nents).
Let
S
⊆
VCN(G)
be
a
nonempty
subset
of
VCN(G).
Then:
Combinatorial
anabelian
topics
I
(i)
41
It
holds
that
Aut
|S|
(G)
=
Aut
Π
z
(G)
z∈S
—
where
we
use
the
notation
Π
z
to
denote
a
VCN-subgroup
[cf.
Definition
2.1,
(i)]
of
Π
G
associated
to
z
∈
VCN(G).
(ii)
It
holds
that
Aut
|grph|
(G)
=
Out
Π
z
(Π
G
)
z∈VCN(G)
—
where
we
use
the
notation
Π
z
to
denote
a
VCN-subgroup
of
Π
G
associated
to
z
∈
VCN(G).
(iii)
The
closed
subgroups
Aut
|S|
(G),
Aut
S
(G)
⊆
Aut(G)
are
open
in
Aut(G).
Moreover,
the
closed
subgroup
Aut
|S|
(G)
⊆
Aut
S
(G)
is
normal
in
Aut
S
(G).
In
particular,
Aut
|grph|
(G)
⊆
Aut(G)
is
normal
in
Aut(G).
Proof.
Assertion
(i)
follows
immediately
from
[CmbGC],
Proposi-
tion
1.2,
(i).
Next,
we
verify
assertion
(ii).
It
follows
immediately
from
[CmbGC],
Proposition
1.5,
(ii),
that
the
right-hand
side
of
the
equality
in
the
statement
of
assertion
(ii)
is
contained
in
Aut(G).
Thus,
asser-
tion
(ii)
follows
immediately
from
assertion
(i).
Assertion
(iii)
follows
immediately
from
the
finiteness
of
the
semi-graph
G,
together
with
the
various
definitions
involved.
Q.E.D.
Definition
2.8
(cf.
the
operation
(Op4)
discussed
at
the
beginning
the
present
§2).
Let
S
⊆
Node(G)
be
a
subset
of
Node(G).
Then
we
define
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
G
S
as
follows:
def
(i)
We
take
Cusp(G
S
)
=
Cusp(G).
(ii)
We
take
Node(G
S
)
=
Node(G)
\
S.
(iii)
We
take
Vert(G
S
)
to
be
the
set
of
connected
components
of
the
semi-graph
obtained
from
G
by
omitting
the
edges
e
∈
Edge(G)
\
S.
Alternatively,
one
may
take
Vert(G
S
)
to
be
def
42
Yuichiro
Hoshi
and
Shinichi
Mochizuki
the
set
of
equivalence
classes
of
elements
of
Vert(G)
with
re-
spect
to
the
equivalence
relation
“∼”
defined
as
follows:
for
v,
w
∈
Vert(G),
v
∼
w
if
either
v
=
w
or
there
exist
n
elements
e
1
,
·
·
·
,
e
n
∈
S
of
S
and
n
+
1
vertices
v
0
,
v
1
,
·
·
·
,
v
n
∈
Vert(G)
def
def
of
G
such
that
v
0
=
v,
v
n
=
w,
and,
for
1
≤
i
≤
n,
it
holds
that
V(e
i
)
=
{v
i−1
,
v
i
}.
(iv)
For
each
branch
b
of
an
edge
e
∈
Edge(G
S
)
(=
Edge(G)
\
S
—
cf.
(i),
(ii))
and
each
vertex
v
∈
Vert(G
S
)
of
G
S
,
b
abuts,
relative
to
G
S
,
to
v
if
b
abuts,
relative
to
G,
to
an
element
of
the
equivalence
class
v
[cf.
(iii)].
(v)
For
each
edge
e
∈
Edge(G
S
)
(=
Edge(G)
\
S
—
cf.
(i),
(ii))
of
G
S
,
we
take
the
anabelioid
of
G
S
corresponding
to
e
∈
Edge(G
S
)
to
be
G
e
[cf.
Definition
2.1,
(ii)].
(vi)
Let
v
∈
Vert(G
S
)
be
a
vertex
of
G
S
.
Then
one
verifies
easily
that
there
exists
a
unique
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
H
v
of
G
such
that
the
set
of
vertices
of
H
v
consists
of
the
elements
of
the
equivalence
class
v
[cf.
(iii)].
Write
def
T
v
=
Node(G|
H
v
)
\
(S
∩
Node(G|
H
v
))
[cf.
Definition
2.2,
(ii)].
Then
we
take
the
anabelioid
of
G
S
corresponding
to
v
∈
Vert(G
S
)
to
be
the
anabelioid
deter-
mined
by
the
finite
étale
coverings
of
(G|
H
v
)
T
v
[cf.
Definition
2.5,
(ii)]
of
degree
a
product
of
primes
∈
Σ.
(vii)
Let
b
be
a
branch
of
an
edge
e
∈
Edge(G
S
)
(=
Edge(G)
\
S
—
cf.
(i),
(ii))
that
abuts
to
a
vertex
v
∈
Vert(G
S
).
Then
since
b
abuts
to
v,
one
verifies
easily
that
there
exists
a
unique
vertex
w
of
G
which
belongs
to
the
equivalent
class
v
[cf.
(iii)]
such
that
b
abuts
to
w
relative
to
G.
We
take
the
morphism
of
anabelioids
associated
to
b,
relative
to
G
S
,
to
be
the
morphism
naturally
determined
by
the
morphism
of
anabelioids
G
e
→
G
w
corresponding
to
the
branch
b
relative
to
G
and
the
morphism
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type
G|
w
→
(G|
H
v
)
T
v
Combinatorial
anabelian
topics
I
43
[cf.
(vi);
Definition
2.1,
(iii)].
Here,
we
recall
that
the
anabe-
lioid
obtained
by
considering
the
connected
finite
étale
cov-
erings
of
G|
w
may
be
naturally
identified
with
G
w
[cf.
Re-
mark
2.1.1].
We
shall
refer
to
this
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
G
S
as
the
generization
of
G
with
respect
to
S
[cf.
Fig.
5
below].
×
G
e
×
⇓
×
×
G
{e}
Figure
5:
Generization
Remark
2.8.1.
It
follows
immediately
from
the
various
definitions
involved
that
if
G
is
of
type
(g,
r)
[cf.
Definition
2.3,
(i)],
then
the
generization
G
Node(G)
of
G
with
respect
to
Node(G)
is
isomorphic
to
model
[cf.
Definition
2.3,
(v)].
G
g,r
Proposition
2.9
(Specialization
outer
isomorphisms).
Let
S
⊆
Node(G)
be
a
subset
of
Node(G).
Write
Π
G
S
for
the
[pro-Σ]
funda-
mental
group
of
the
generization
G
S
of
G
with
respect
to
S
[cf.
Defi-
nition
2.8].
Then
the
following
hold:
44
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(i)
There
exists
a
natural
outer
isomorphism
of
profinite
groups
∼
Φ
G
S
:
Π
G
S
−→
Π
G
which
satisfies
the
following
three
conditions:
(1)
Φ
G
S
induces
a
bijection
between
the
set
of
cuspidal
sub-
groups
of
Π
G
S
and
the
set
of
cuspidal
subgroups
of
Π
G
.
(2)
Φ
G
S
induces
a
bijection
between
the
set
of
nodal
sub-
groups
of
Π
G
S
and
the
set
of
nodal
subgroups
of
Π
G
as-
sociated
to
the
elements
of
Node(G)
\
S.
(3)
Let
v
∈
Vert(G
S
)
be
a
vertex
of
G
S
;
H
v
,
T
v
as
in
Definition
2.8,
(vi).
Then
Φ
G
S
induces
a
bijection
be-
tween
the
Π
G
S
-conjugacy
class
of
any
verticial
subgroup
Π
v
⊆
Π
G
S
of
Π
G
S
associated
to
v
∈
Vert(G
S
)
and
the
Π
G
-conjugacy
class
of
subgroups
determined
by
the
image
of
the
outer
homomorphism
Π
(G|
H
v
)
Tv
−→
Π
G
induced
by
the
natural
morphism
(G|
H
v
)
T
v
→
G
[cf.
Def-
initions
2.2,
(ii);
2.5,
(ii)]
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type.
∼
Moreover,
any
two
outer
isomorphisms
Π
G
S
→
Π
G
that
sat-
isfy
the
above
three
conditions
differ
by
composition
with
a
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)]
outomorphism
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
of
Π
G
S
.
(ii)
The
isomorphism
∼
Out(Π
G
)
−→
Out(Π
G
S
)
induced
by
the
natural
outer
isomorphism
of
(i)
determines
an
injection
Aut
S
(G)
→
Aut(G
S
)
[cf.
Definition
2.6,
(i)].
Proof.
First,
we
verify
assertion
(i).
An
outer
isomorphism
that
satisfies
the
three
conditions
of
assertion
(i)
may
be
obtained
by
ob-
serving
that,
after
sorting
through
the
various
definitions
involved,
a
finite
étale
covering
of
G
S
amounts
to
the
same
data
as
a
finite
étale
covering
of
G.
The
final
portion
of
assertion
(i)
follows
immediately,
in
light
of
the
three
conditions
in
the
statement
of
assertion
(i),
from
Combinatorial
anabelian
topics
I
45
[CmbGC],
Proposition
1.5,
(ii).
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
[CmbGC],
Proposition
1.5,
(ii),
together
with
the
three
conditions
in
the
statement
of
assertion
(i).
This
completes
the
proof
of
Proposition
2.9.
Q.E.D.
Definition
2.10.
Let
S
⊆
Node(G)
be
a
subset
of
Node(G).
Write
Π
G
S
for
the
[pro-Σ]
fundamental
group
of
the
generization
G
S
of
G
with
respect
to
S
[cf.
Definition
2.8].
Then
we
shall
refer
to
the
natural
outer
isomorphism
∼
Φ
G
S
:
Π
G
S
−→
Π
G
obtained
in
Proposition
2.9,
(i),
as
the
specialization
outer
isomorphism
with
respect
to
S.
Proposition
2.11
(Commensurable
terminality
of
closed
subgroups
determined
by
certain
semi-graphs).
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G
and
S
⊆
Node(G|
H
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
separating
type
[cf.
Definition
2.5,
(i)].
Then
the
natural
morphism
(G|
H
)
S
→
G
[cf.
Definitions
2.2,
(ii);
2.5,
(ii)]
of
semi-graphs
of
an-
abelioids
of
pro-Σ
PSC-type
determines
an
outer
injection
of
profinite
groups
Π
(G|
H
)
S
→
Π
G
.
Moreover,
the
image
of
this
outer
injection
is
commensurably
termi-
nal
in
Π
G
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
def
def
Proof.
Write
H
=
(G|
H
)
S
and
T
=
Node(G|
H
)
\
S.
Note
that
it
follows
from
the
definition
of
G|
H
that
T
may
be
regarded
as
the
subset
of
Node(G)
determined
by
Node(H);
for
simplicity,
we
shall
identify
T
with
Node(H).
Now
it
follows
immediately
from
the
definition
of
“G
T
”
that
the
composite
Φ
H,S
Φ
−1
G
T
∼
Π
H
−→
Π
G
−→
Π
G
T
factors
through
a
verticial
subgroup
Π
v
⊆
Π
G
T
of
Π
G
T
associated
to
a
vertex
v
∈
Vert(G
T
),
and
that
the
composite
Π
H
−→
Π
(G
T
)|
v
of
the
resulting
outer
homomorphism
Π
H
→
Π
v
[which
is
well-defined
in
light
of
the
commensurable
terminality
of
Π
v
in
Π
G
S
—
cf.
[CmbGC],
46
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Proposition
1.2,
(ii)]
and
the
natural
outer
isomorphism
Π
v
≃
Π
(G
T
)|
v
[cf.
Remark
2.1.1]
may
be
identified
with
“Φ
−1
H
T
”
[cf.
Definition
2.10].
Thus,
Proposition
2.11
follows
immediately
from
the
fact
that
Φ
H
T
is
an
outer
isomorphism,
together
with
the
fact
that
Π
v
⊆
Π
G
S
is
commensurably
terminal
in
Π
G
S
[cf.
[CmbGC],
Proposition
1.2,
(ii)].
This
completes
the
proof
of
Proposition
2.11.
Q.E.D.
Lemma
2.12
(Restrictions
of
outomorphisms).
Let
H
⊆
Π
G
be
a
closed
subgroup
of
Π
G
which
is
normally
terminal
[cf.
the
discus-
sion
entitled
“Topological
groups”
in
§0]
and
α
∈
Out
H
(Π
G
)
[cf.
Defini-
tion
2.6,
(ii)].
Then
the
following
hold:
(i)
preserves
There
exists
a
lifting
α
∈
Aut(Π
G
)
of
α
such
that
α
is
the
closed
subgroup
H
⊆
Π
G
.
Moreover,
such
a
lifting
α
uniquely
determined
up
to
composition
with
an
H-inner
automorphism
of
Π
G
.
(ii)
Write
α
H
for
the
outomorphism
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
of
H
determined
by
the
restriction
of
a
lifting
α
as
obtained
in
(i)
to
the
closed
subgroup
H
⊆
Π
G
.
Then
the
map
Out
H
(Π
G
)
−→
Out(H)
given
by
assigning
α
→
α
H
is
a
homomorphism.
(iii)
The
homomorphism
Out
H
(Π
G
)
−→
Out(H)
obtained
in
(ii)
depends
only
on
the
conjugacy
class
of
the
def
closed
subgroup
H
⊆
Π
G
,
i.e.,
if
we
write
H
γ
=
γ
·
H
·
γ
−1
for
γ
∈
Π
G
,
then
the
diagram
Out
H
(Π
G
)
−−−−→
Out(H)
⏐
⏐
γ
Out
H
(Π
G
)
−−−−→
Out(H
γ
)
—
where
the
upper
(respectively,
lower)
horizontal
arrow
is
the
homomorphism
given
by
mapping
α
→
α
H
(respectively,
α
→
α
H
γ
),
and
the
right-hand
vertical
arrow
is
the
isomorphism
Combinatorial
anabelian
topics
I
47
obtained
by
mapping
φ
∈
Out(H)
to
H
γ
Inn(γ
−1
)
∼
−→
φ
∼
Inn(γ)
∼
H
−→
H
−→
H
γ
—
commutes.
Proof.
Assertion
(i)
follows
immediately
from
the
normal
termi-
nality
of
H
in
Π
G
.
Assertion
(ii)
follows
immediately
from
assertion
(i).
Assertion
(iii)
follows
immediately
from
the
various
definitions
in-
volved.
Q.E.D.
Definition
2.13.
Let
H
⊆
Π
G
be
a
[closed]
subgroup
of
Π
G
which
is
normally
terminal
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Then
we
shall
write
Out
|H|
(Π
G
)
⊆
Out
H
(Π
G
)
for
the
closed
subgroup
of
Out
H
(Π
G
)
consisting
of
outomorphisms
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
α
of
Π
G
such
that
the
image
α
H
of
α
via
the
homomorphism
Out
H
(G)
→
Out(H)
obtained
in
Lemma
2.12,
(ii),
is
trivial.
Also,
we
shall
write
Aut
|H|
(G)
=
Out
|H|
(Π
G
)
∩
Aut(G)
.
def
Definition
2.14.
(i)
Let
T
⊆
Cusp(G)
be
an
omittable
[cf.
Definition
2.4,
(i)]
sub-
set
of
Cusp(G).
Write
Π
G
•T
for
the
[pro-Σ]
fundamental
group
of
G
•T
[cf.
Definition
2.4,
(ii)]
and
N
T
⊆
Π
G
for
the
nor-
mal
closed
subgroup
of
Π
G
topologically
normally
generated
by
the
cuspidal
subgroups
of
Π
G
associated
to
elements
of
T
.
Then
one
verifies
easily
that
the
natural
outer
isomorphism
∼
Π
G
/N
T
→
Π
G
•T
[cf.
Definition
2.4,
(ii)]
induces
a
homomor-
phism
Out
N
T
(Π
G
)
→
Out(Π
G
•T
)
that
fits
into
a
commutative
diagram
Aut
T
(G)
⏐
⏐
−−−−→
Aut(G
•T
)
⏐
⏐
Out
N
T
(Π
G
)
−−−−→
Out(Π
G
•T
)
48
Yuichiro
Hoshi
and
Shinichi
Mochizuki
—
where
the
vertical
arrows
are
the
natural
injections.
For
α
∈
Out
N
T
(Π
G
),
we
shall
write
α
G
•T
∈
Out(Π
G
•T
)
for
the
image
of
α
via
the
lower
horizontal
arrow
in
the
above
commutative
diagram.
If,
moreover,
α
∈
Aut
T
(G),
then,
in
light
of
the
injectivity
of
the
right-hand
vertical
arrow
in
the
above
diagram,
we
shall
write
[by
abuse
of
notation]
α
G
•T
∈
Aut(G
•T
)
for
the
image
of
α
via
the
upper
horizontal
arrow
in
the
above
commutative
diagram.
(ii)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G
and
S
⊆
Node(G|
H
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
separating
type
[cf.
Definition
2.5,
(i)].
Write
Π
(G|
H
)
S
for
the
[pro-Σ]
fundamental
group
of
(G|
H
)
S
[cf.
Definition
2.5,
(ii)].
Then
the
natural
outer
homomor-
phism
Π
(G|
H
)
S
→
Π
G
is
an
outer
injection
whose
image
is
commensurably
terminal
[cf.
Proposition
2.11].
Thus,
it
fol-
lows
from
Lemma
2.12,
(iii),
that
we
have
a
homomorphism
Out
Π
(G|
H
)
S
(Π
G
)
→
Out(Π
(G|
H
)
S
)
that
fits
into
a
commuta-
tive
diagram
Aut
HS
(G)
=
Aut
H
(G)
∩
Aut
S
(G)
−−−−→
Aut((G|
H
)
S
)
⏐
⏐
⏐
⏐
def
Out
Π
(G|
H
)
S
(Π
G
)
−−−−→
Out(Π
(G|
H
)
S
)
—
where
the
vertical
arrows
are
the
natural
injections.
For
α
∈
Out
Π
(G|
H
)
S
(Π
G
),
we
shall
write
α
(G|
H
)
S
∈
Out(Π
(G|
H
)
S
)
for
the
image
of
α
via
the
lower
horizontal
arrow
in
the
above
commutative
diagram.
If,
moreover,
α
∈
Aut
HS
(G),
then,
in
light
of
the
injectivity
of
the
right-hand
vertical
arrow
in
the
above
diagram,
we
shall
write
[by
abuse
of
notation]
α
(G|
H
)
S
∈
Aut((G|
H
)
S
)
for
the
image
of
α
via
the
upper
horizontal
arrow
in
the
above
commutative
diagram.
Finally,
if
T
⊆
Cusp((G|
H
)
S
)
is
an
Combinatorial
anabelian
topics
I
49
omittable
subset
of
Cusp((G|
H
)
S
),
then
we
shall
write
Aut
HS•T
(G)
⊆
Aut
HS
(G)
for
the
inverse
image
of
the
closed
subgroup
Aut
T
((G|
H
)
S
)
⊆
Aut((G|
H
)
S
)
of
Aut((G|
H
)
S
)
in
Aut
HS
(G)
via
the
upper
horizontal
arrow
Aut
HS
(G)
→
Aut((G|
H
)
S
)
of
the
above
commutative
diagram;
thus,
we
have
a
natural
homomorphism
[cf.
(i)]
Aut
HS•T
(G)
α
(iii)
−→
Aut(((G|
H
)
S
)
•T
)
→
α
((G|
H
)
S
)
•T
.
Let
z
∈
VCN(G)
be
an
element
of
VCN(G)
and
Π
z
⊆
Π
G
a
VCN-subgroup
of
Π
G
associated
to
z
∈
VCN(G).
Then
it
follows
from
[CmbGC],
Proposition
1.2,
(ii),
that
the
closed
subgroup
Π
z
⊆
Π
G
is
commensurably
terminal.
Thus,
it
fol-
lows
from
Lemma
2.12,
(iii),
that
we
obtain
a
homomorphism
Out
Π
z
(Π
G
)
→
Out(Π
z
)
that
fits
into
a
commutative
diagram
Aut
{z}
(G)
−−−−→
Aut(G
z
)
⏐
⏐
⏐
⏐
Out
Π
z
(Π
G
)
−−−−→
Out(Π
z
)
—
where
the
left-hand
vertical
arrow
is
injective,
and
the
right-
hand
vertical
arrow
is
an
isomorphism.
For
α
∈
Out
Π
z
(Π
G
),
we
shall
write
α
z
∈
Out(Π
z
)
for
the
image
of
α
via
the
lower
horizontal
arrow
in
the
above
commutative
diagram.
§3.
Synchronization
of
cyclotomes
In
the
present
§,
we
introduce
and
study
the
notion
of
the
second
cohomology
group
with
compact
supports
of
a
semi-graph
of
anabelioids
of
PSC-type
[cf.
Definition
3.1,
(ii),
(iii)
below].
In
particular,
we
show
that
such
cohomology
groups
are
compatible
with
graph-theoretic
lo-
calization
[cf.
Definition
3.4,
Lemma
3.5
below].
This
leads
naturally
50
Yuichiro
Hoshi
and
Shinichi
Mochizuki
to
a
discussion
of
the
phenomenon
of
synchronization
among
the
vari-
ous
cyclotomes
[cf.
Definition
3.8
below]
arising
from
a
semi-graph
of
anabelioids
of
PSC-type
[cf.
Corollary
3.9
below].
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Write
G
for
the
underlying
semi-graph
of
G,
Π
G
for
the
[pro-Σ]
fundamental
group
of
G,
and
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Σ
-module
and
v
∈
Definition
3.1.
Let
M
be
a
finitely
generated
Z
Vert(G)
a
vertex
of
G.
(i)
We
shall
write
def
H
2
(G,
M
)
=
H
2
(Π
G
,
M
)
—
where
we
regard
M
as
being
equipped
with
the
trivial
action
of
Π
G
—
and
refer
to
H
2
(G,
M
)
as
the
second
cohomology
group
of
G.
(ii)
Let
s
be
a
section
of
the
natural
surjection
Cusp(
G)
Cusp(G).
Given
a
central
extension
of
profinite
groups
1
−→
M
−→
E
−→
Π
G
−→
1
,
and
a
cusp
e
∈
Cusp(G),
we
shall
refer
to
a
section
of
this
extension
over
the
edge-like
subgroup
Π
s(e)
⊆
Π
G
of
Π
G
deter-
as
a
trivialization
of
this
extension
mined
by
s(e)
∈
Cusp(
G)
at
the
cusp
e.
We
shall
write
H
c
2
(G,
M
)
for
the
set
of
equivalence
classes
[E,
(ι
e
:
Π
s(e)
→
E)
e∈Cusp(G)
]
of
collections
of
data
(E,
(ι
e
:
Π
s(e)
→
E)
e∈Cusp(G)
)
as
follows:
(a)
E
is
a
central
extension
of
profinite
groups
1
−→
M
−→
E
−→
Π
G
−→
1
;
(b)
for
each
e
∈
Cusp(G),
ι
e
is
a
trivialization
of
this
extension
at
the
cusp
e.
The
equivalence
relation
“∼”
is
then
defined
as
follows:
for
two
collections
of
data
(E,
(ι
e
))
and
(E
,
(ι
e
)),
we
shall
write
(E,
(ι
e
))
∼
(E
,
(ι
e
))
if
there
exists
an
isomorphism
Combinatorial
anabelian
topics
I
51
∼
of
profinite
groups
α
:
E
→
E
over
Π
G
which
induces
the
iden-
tity
automorphism
of
M
,
and,
moreover,
for
each
e
∈
Cusp(G),
maps
ι
e
to
ι
e
.
We
shall
refer
to
H
c
2
(G,
M
)
as
the
second
coho-
mology
group
with
compact
supports
of
G.
(iii)
We
shall
write
def
H
c
2
(v,
M
)
=
H
c
2
(G|
v
,
M
)
[cf.
(ii);
Definition
2.1,
(iii)]
and
refer
to
H
c
2
(v,
M
)
as
the
second
cohomology
group
with
compact
supports
of
v.
(iv)
Σ
-
The
set
H
c
2
(G,
M
)
is
equipped
with
a
natural
structure
of
Z
module
defined
as
follows:
•
Let
[E,
(ι
e
)],
[E
,
(ι
e
)]
∈
H
c
2
(G,
M
).
Then
the
fiber
prod-
uct
E
×
Π
G
E
of
the
surjections
E
Π
G
,
E
Π
G
is
an
extension
of
Π
G
by
M
×
M
.
Thus,
the
quotient
S
of
E
×
Π
G
E
by
the
image
of
the
composite
M
m
→
→
M
×
M
(m,
−m)
→
E
×
Π
G
E
is
an
extension
of
Π
G
by
M
.
On
the
other
hand,
it
follows
from
the
definition
of
S
that
for
each
e
∈
Cusp(G),
the
sec-
tions
ι
e
and
ι
e
naturally
determine
a
section
ι
Se
:
Π
s(e)
→
S
over
Π
s(e)
.
Thus,
we
define
[E,
(ι
e
)]
+
[E
,
(ι
e
)]
=
[S,
(ι
Se
)]
.
def
Here,
one
may
verify
easily
that
the
equivalence
class
[S,
(ι
Se
)]
depends
only
on
the
equivalence
classes
[E,
(ι
e
)],
[E
,
(ι
e
)],
and
that
this
definition
of
“+”
determines
a
module
structure
on
H
c
2
(G,
M
).
•
Let
[E,
(ι
e
)]
∈
H
c
2
(G,
M
)
be
an
element
of
H
c
2
(G,
M
)
and
pr
Σ
.
Now
the
composite
E
×
M
1
E
Π
G
deter-
a
∈
Z
mines
an
extension
of
Π
G
by
M
×
M
.
Thus,
the
quotient
P
of
E
×
M
by
the
image
of
the
composite
M
m
→
→
M
×
M
(m,
−am)
→
E
×
M
is
an
extension
of
Π
G
by
M
.
On
the
other
hand,
it
follows
from
the
definition
of
P
that
for
each
e
∈
Cusp(G),
the
52
Yuichiro
Hoshi
and
Shinichi
Mochizuki
section
ι
e
and
the
zero
homomorphism
Π
s(e)
→
M
natu-
rally
determine
a
section
ι
P
e
:
Π
s(e)
→
P
over
Π
s(e)
.
Thus,
we
define
def
a
·
[E,
(ι
e
)]
=
[P,
(ι
P
e
)]
.
Here,
one
may
verify
easily
that
the
equivalence
class
[P,
(ι
P
e
)]
depends
only
on
the
equivalence
class
[E,
(ι
e
)]
Σ
,
and
that
this
definition
of
“·”
determines
a
and
a
∈
Z
Σ
Z
-module
structure
on
H
c
2
(G,
M
).
Finally,
we
note
that
it
follows
from
Lemma
3.2
below
that
Σ
-module
“H
2
(G,
M
)”
does
not
depend
on
the
choice
of
the
Z
c
Σ
-module
“H
2
(G,
M
)”
is
the
section
s.
More
precisely,
the
Z
c
uniquely
determined
by
G
and
M
up
to
the
natural
isomorphism
obtained
in
Lemma
3.2.
Lemma
3.2
(Independence
of
the
choice
of
section).
Let
M
Σ
-module
and
s,
s
sections
of
the
natural
sur-
be
a
finitely
generated
Z
Cusp(G).
Write
H
c
2
(G,
M,
s),
H
c
2
(G,
M,
s
)
for
the
jection
Cusp(
G)
Σ
-modules
“H
2
(G,
M
)”
defined
in
Definition
3.1
by
means
of
the
sec-
Z
c
tions
s,
s
,
respectively.
Then
there
exists
a
natural
isomorphism
of
Σ
-modules
Z
∼
H
c
2
(G,
M,
s)
−→
H
c
2
(G,
M,
s
)
.
Proof.
Let
[E,
(ι
e
)]
∈
H
c
2
(G,
M,
s)
be
an
element
of
H
c
2
(G,
M,
s).
Now
it
follows
from
the
various
definitions
involved
that,
for
each
e
∈
Cusp(G),
there
exists
an
element
γ
e
∈
Π
G
such
that
Π
s
(e)
=
γ
e
·
Π
s(e)
·
γ
e
−1
.
For
each
e
∈
Cusp(G),
fix
a
lifting
γ
e
∈
E
of
γ
e
∈
Π
G
and
write
ι
e
:
Π
s
(e)
→
E
for
the
section
given
by
Π
s
(e)
=
γ
e
·
Π
s(e)
·
γ
e
−1
γ
e
aγ
e
−1
−→
E
→
γ
e
ι
e
(a)
γ
e
−1
.
Then
it
follows
immediately
from
the
fact
that
M
⊆
E
is
contained
in
the
center
Z(E)
of
E
that
this
section
ι
e
does
not
depend
on
the
choice
of
the
lifting
γ
e
∈
E
of
γ
e
∈
Π
G
.
Moreover,
it
follows
immediately
from
the
various
definitions
involved
that
the
assignment
“[E,
(ι
e
)]
→
[E,
(ι
e
)]”
Σ
-modules
determines
an
isomorphism
of
Z
∼
H
c
2
(G,
M,
s)
−→
H
c
2
(G,
M,
s
)
.
This
completes
the
proof
of
Lemma
3.2.
Q.E.D.
Combinatorial
anabelian
topics
I
53
Lemma
3.3
(Exactness
of
certain
sequences).
Let
M
be
a
Σ
-module.
Suppose
that
Cusp(G)
=
∅.
Then
the
finitely
generated
Z
natural
inclusions
Π
e
→
Π
G
—
where
e
ranges
over
the
cusps
of
G,
and,
for
each
cusp
e
∈
Cusp(G),
we
use
the
notation
Π
e
to
denote
an
edge-like
subgroup
of
Π
G
associated
to
the
cusp
e
—
determine
an
exact
sequence
Σ
-modules
of
Z
Hom
Z
Σ
(Π
ab
Hom
Z
Σ
(Π
e
,
M
)
−→
H
c
2
(G,
M
)
−→
0
.
G
,
M
)
−→
e∈Cusp(G)
Proof.
Let
s
be
a
section
of
the
natural
surjection
Cusp(
G)
Cusp(G).
Then
given
an
element
Hom
Z
Σ
(Π
e
,
M
)
,
(φ
e
:
Π
e
→
M
)
e∈Node(G)
∈
e∈Cusp(G)
one
may
construct
an
element
pr
2
[M
×
Π
G
(
Π
G
),
(ι
e
:
Π
s(e)
→
M
×
Π
G
)
e∈Node(G)
]
—
where
we
write
ι
e
:
Π
s(e)
→
M
×
Π
G
for
the
section
determined
by
φ
e
:
Π
s(e)
→
M
and
the
natural
inclusion
Π
s(e)
→
Π
G
—
of
H
c
2
(G,
M
).
In
particular,
we
obtain
a
map
e∈Cusp(G)
Hom
Z
Σ
(Π
e
,
M
)
→
H
c
2
(G,
M
),
Σ
-modules.
Now
the
which,
as
is
easily
verified,
is
a
homomorphism
of
Z
exactness
of
the
sequence
in
question
follows
immediately
from
the
fact
that
Π
G
is
free
pro-Σ
[cf.
[CmbGC],
Remark
1.1.3].
This
completes
the
proof
of
Lemma
3.3.
Q.E.D.
Σ
-module.
Definition
3.4.
Let
M
be
a
finitely
generated
Z
(i)
Let
E
be
a
semi-graph
of
anabelioids.
Denote
by
VCN(E)
the
set
of
components
of
E
[i.e.,
the
set
of
vertices
and
edges
of
E]
and,
for
each
z
∈
VCN(E),
by
Π
E
z
the
fundamental
group
of
the
anabelioid
E
z
of
E
corresponding
to
z
∈
VCN(E).
Then
we
define
a
central
extension
of
G
by
M
to
be
a
collection
of
data
∼
(E,
α
=
(α
z
:
M
→
Π
E
z
)
z∈VCN(E)
,
β
:
E/α
→
G)
as
follows:
(a)
For
each
z
∈
VCN(E),
α
z
:
M
→
Π
E
z
is
an
injective
ho-
momorphism
of
profinite
groups
whose
image
is
contained
54
Yuichiro
Hoshi
and
Shinichi
Mochizuki
in
the
center
Z(Π
E
z
)
of
Π
E
z
.
[Thus,
the
image
of
α
z
is
a
normal
closed
subgroup
of
Π
E
z
.]
(b)
For
each
branch
b
of
an
edge
e
that
abuts
to
a
vertex
v
of
E,
we
assume
that
the
outer
homomorphism
Π
E
e
→
Π
E
v
associated
to
b
is
injective
and
fits
into
a
commutative
diagram
of
[outer]
homomorphisms
of
profinite
groups
M
⏐
⏐
α
v
M
⏐
⏐
α
e
Π
E
e
−−−−→
Π
E
v
—
i.e.,
where
the
lower
horizontal
arrow
is
the
outer
in-
jection
associated
to
b.
(c)
Write
E/α
for
the
semi-graph
of
anabelioids
defined
as
follows:
We
take
the
underlying
semi-graph
of
E/α
to
be
the
underlying
semi-graph
of
E;
for
each
z
∈
VCN(E),
we
take
the
anabelioid
(E/α)
z
of
E/α
corresponding
to
z
∈
VCN(E)
to
be
the
anabelioid
determined
by
the
profinite
group
Π
E
z
/Im(α
z
)
[cf.
condition
(a)];
for
each
branch
b
of
an
edge
e
that
abuts
to
a
vertex
v
of
E,
we
take
the
associated
morphism
of
anabelioids
(E/α)
e
→
(E/α)
v
to
be
the
morphism
of
anabelioids
naturally
determined
by
the
morphism
E
e
→
E
v
associated,
relative
to
E,
to
b
[cf.
condition
(b)].
(d)
β
:
E/α
→
G
is
an
isomorphism
of
semi-graphs
of
anabe-
lioids.
∼
There
is
an
evident
notion
of
isomorphisms
of
central
exten-
sions
of
G
by
M
.
Also,
given
a
central
extension
of
G
by
M
,
Cusp(G),
and
a
section
s
of
the
natural
surjection
Cusp(
G)
there
is
an
evident
notion
of
trivialization
of
the
given
cen-
tral
extension
of
G
by
M
at
a
cusp
of
G
[cf.
the
discussion
of
Definition
3.1,
(ii),
(iv)].
(ii)
Let
1
−→
M
−→
E
−→
Π
G
−→
1
be
a
central
extension
of
Π
G
by
M
.
Then
we
shall
define
a
semi-graph
of
anabelioids
G
E
Combinatorial
anabelian
topics
I
55
—
which
we
shall
refer
to
as
the
semi-graph
of
anabelioids
associated
to
the
central
extension
E
—
as
follows:
We
take
the
underlying
semi-graph
of
G
E
to
be
the
underlying
semi-graph
of
G.
We
take
the
anabelioid
of
G
E
corresponding
to
z
∈
VCN(G)
to
be
the
anabelioid
determined
by
the
fiber
product
E
×
Π
G
Π
z
of
the
surjection
E
→
Π
G
and
a
natural
inclusion
Π
z
→
Π
G
—
where
we
use
the
notation
Π
z
⊆
Π
G
to
denote
a
VCN-subgroup
[cf.
Definition
2.1,
(i)]
of
Π
G
associated
to
z
∈
VCN(G);
for
each
branch
b
of
an
edge
e
that
abuts
to
a
vertex
v
of
G,
if
we
write
(G
E
)
v
,
(G
E
)
e
for
the
anabelioids
of
G
E
corresponding
to
v,
e,
respectively,
then
we
take
the
morphism
of
anabelioids
(G
E
)
e
→
(G
E
)
v
associated
to
the
branch
b
to
be
the
morphism
naturally
determined
by
the
morphism
of
anabelioids
G
e
→
G
v
associated,
relative
to
G,
to
b.
(iii)
In
the
notation
of
(ii),
one
may
verify
easily
that
the
semi-
graph
of
anabelioids
G
E
associated
to
the
central
extension
E
is
equipped
with
a
natural
structure
of
central
extension
of
G
by
M
.
More
precisely,
for
each
z
∈
VCN(G),
if
we
denote
by
α
z
:
M
→
Π
(G
E
)
z
=
E
×
Π
G
Π
z
the
homomorphism
deter-
mined
by
the
natural
inclusion
M
→
E
and
the
trivial
homo-
morphism
M
→
Π
z
,
then
there
exists
a
natural
isomorphism
∼
β
:
G
E
/(α
z
)
z∈VCN(G)
→
G
such
that
the
collection
of
data
(G
E
,
(α
z
)
z∈VCN(G)
,
β)
forms
a
central
extension
of
G
by
M
,
which
we
shall
refer
to
as
the
central
extension
of
G
by
M
associated
to
the
central
extension
E.
Lemma
3.5
(Graph-theoretic
localizability
of
central
exten-
Σ
-
sions
of
fundamental
groups).
Let
M
be
a
finitely
generated
Z
module.
Then
the
following
hold:
(i)
(Exactness
and
centrality)
Let
∼
(E,
α
=
(α
z
:
M
→
Π
E
z
)
z∈VCN(E)
,
β
:
E/α
→
G)
(‡
1
)
56
Yuichiro
Hoshi
and
Shinichi
Mochizuki
be
a
central
extension
of
G
by
M
[cf.
Definition
3.4,
(i)].
Write
Π
E
for
the
pro-Σ
fundamental
group
of
E,
i.e.,
the
max-
imal
pro-Σ
quotient
of
the
fundamental
group
of
E
[cf.
the
dis-
cussion
preceding
[SemiAn],
Definition
2.2].
Then
the
compos-
β
∼
ite
E
→
E/α
→
G
determines
an
exact
sequence
of
profinite
groups
1
−→
M
−→
Π
E
−→
Π
G
−→
1
(‡
2
)
which
is
central.
(ii)
(Natural
isomorphism
I)
In
the
notation
of
(i),
the
central
extension
of
G
by
M
associated
to
the
central
extension
(‡
2
)
[cf.
Definition
3.4,
(iii)]
is
naturally
isomorphic,
as
a
central
extension
of
G
by
M
,
to
(‡
1
).
(iii)
(Natural
isomorphism
II)
Let
1
−→
M
−→
E
−→
Π
G
−→
1
be
a
central
extension
of
Π
G
by
M
.
Then
the
pro-Σ
fun-
damental
group
of
the
semi-graph
of
anabelioids
G
E
associated
to
the
central
extension
E
[cf.
Definition
3.4,
(ii)]
—
i.e.,
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
G
E
—
is
naturally
isomorphic,
over
Π
G
,
to
E.
(iv)
(Equivalence
of
categories)
The
correspondences
of
(i),
(ii),
(iii)
determine
a
natural
equivalence
of
categories
between
the
category
of
central
extensions
of
G
by
M
and
the
category
of
central
extensions
of
Π
G
by
M
.
[Here,
we
take
the
morphisms
in
both
categories
to
be
the
isomorphisms
of
central
exten-
sions
of
the
sort
under
consideration.]
Moreover,
this
equiva-
lence
extends
to
a
similar
natural
equivalence
of
categories
between
categories
of
central
extensions
equipped
with
trivial-
izations
at
the
cusps
of
G
[cf.
Definitions
3.1,
(ii);
3.4,
(i)].
Proof.
First,
we
verify
assertion
(i).
If
Node(G)
=
∅,
then
assertion
(i)
is
immediate;
thus,
suppose
that
Node(G)
=
∅.
For
each
connected
finite
étale
covering
E
→
E
of
E,
denote
by
Π
E
the
pro-Σ
fundamental
group
of
E
,
by
VCN(E
)
the
set
of
components
of
E
[i.e.,
the
set
of
vertices
and
edges
of
E
],
and
by
Vert(E
)
the
set
of
vertices
of
E
;
for
each
z
∈
VCN(E
),
denote
by
E
z
the
anabelioid
of
E
corresponding
to
z
∈
VCN(E
)
and
by
Π
E
z
the
fundamental
group
of
E
z
.
Now
we
claim
that
Combinatorial
anabelian
topics
I
57
∼
(∗
1
):
the
composite
in
question
E
→
E/α
→
G
in-
duces
an
isomorphism
between
the
underlying
semi-
graphs,
as
well
as
an
outer
surjection
Π
E
Π
G
.
Indeed,
the
fact
that
the
composite
in
question
determines
an
isomor-
phism
between
the
underlying
semi-graphs
follows
from
conditions
(c),
∼
(d)
of
Definition
3.4,
(i).
In
particular,
we
obtain
a
bijection
VCN(E)
→
∼
VCN(G).
Now
for
each
z
∈
VCN(E)
→
VCN(G),
again
by
conditions
∼
(c),
(d)
of
Definition
3.4,
(i),
the
composite
E
→
E/α
→
G
induces
an
outer
surjection
Π
E
z
Π
z
,
where
we
use
the
notation
Π
z
⊆
Π
G
to
denote
a
VCN-subgroup
[cf.
Definition
2.1,
(i)]
of
Π
G
associated
to
z
∈
VCN(G).
Therefore,
in
light
of
the
isomorphism
verified
above
be-
tween
the
semi-graphs
of
E
and
G,
one
may
verify
easily
that
the
natural
outer
homomorphism
Π
E
→
Π
G
is
surjective.
This
completes
the
proof
of
the
claim
(∗
1
).
∼
For
each
vertex
v
∈
Vert(E)
→
Vert(G)
[cf.
claim
(∗
1
)],
it
follows
from
the
assumption
that
Node(G)
=
∅
that
any
verticial
subgroup
Π
v
⊆
Π
G
of
Π
G
associated
to
a
vertex
v
∈
Vert(G)
is
a
free
pro-Σ
group
[cf.
[CmbGC],
Remark
1.1.3];
thus,
there
exists
a
section
of
the
natural
surjection
Π
E
v
Π
v
.
Now
for
each
vertex
v
∈
Vert(G),
let
us
fix
such
a
section
of
the
natural
surjection
Π
E
v
Π
v
,
hence
also
—
since
the
extension
Π
E
v
of
Π
v
by
M
is
central
[cf.
condition
(a)
of
∼
Definition
3.4,
(i)]
—
an
isomorphism
t
v
:
M
×
Π
v
→
Π
E
v
.
Let
G
1
→
G
def
be
a
connected
finite
étale
Galois
covering
of
G
and
write
E
1
=
E
×
G
G
1
.
Then
it
follows
from
the
claim
(∗
1
)
that
E
1
is
connected;
moreover,
one
may
verify
easily
that
the
structure
of
central
extension
of
G
by
M
on
E
naturally
determines
a
structure
of
central
extension
of
G
1
by
∼
M
on
E
1
,
and
that
for
each
vertex
v
∈
Vert(E)
→
Vert(G)
and
each
∼
vertex
w
∈
Vert(E
1
)
→
Vert(G
1
)
that
lies
over
v,
the
normal
closed
subgroup
Π
(E
1
)
w
⊆
Π
E
v
corresponds
to
M
×
Π
w
⊆
M
×
Π
v
relative
∼
to
the
isomorphism
t
v
:
M
×
Π
v
→
Π
E
v
fixed
above,
i.e.,
we
obtain
an
∼
isomorphism
t
w
:
M
×
Π
w
→
Π
(E
1
)
w
.
Now
for
a
finite
quotient
M
Q
of
M
and
a
connected
finite
étale
Galois
covering
G
1
→
G
of
G,
we
shall
say
that
a
connected
finite
étale
covering
E
2
→
E
of
E
satisfies
the
condition
(†
Q,G
1
)
if
the
following
two
conditions
are
satisfied:
def
(†
1
Q,G
1
)
E
2
→
E
factors
through
E
1
=
E
×
G
G
1
→
E,
the
resulting
covering
E
2
→
E
1
is
Galois,
and
for
each
vertex
v
∈
VCN(E
1
),
the
composite
M
→
Π
(E
1
)
v
→
Π
E
1
Π
E
1
/Π
E
2
58
Yuichiro
Hoshi
and
Shinichi
Mochizuki
is
surjective,
with
kernel
equal
to
the
kernel
of
M
Q.
(†
2
Q,G
1
)
E
2
→
E
is
Galois.
Then
we
claim
that
(∗
2
):
for
any
finite
quotient
M
Q
of
M
and
any
connected
finite
étale
Galois
covering
G
1
→
G,
there
exists
—
after
possibly
replacing
G
1
→
G
by
a
con-
nected
finite
étale
Galois
covering
of
G
that
factors
through
G
1
→
G
—
a
connected
finite
étale
covering
of
E
which
satisfies
the
condition
(†
Q,G
1
).
Indeed,
let
M
Q
be
a
finite
quotient
of
M
,
G
1
→
G
a
connected
finite
def
étale
Galois
covering
of
G,
and
E
1
=
E
×
G
G
1
.
For
each
vertex
v
∈
∼
Vert(E
1
)
→
Vert(G
1
)
[cf.
the
above
discussion],
denote
by
Π
(E
1
)
v
Q
v
the
quotient
of
Π
(E
1
)
v
obtained
by
forming
the
composite
t
v
∼
pr
1
Π
(E
1
)
v
←
M
×
Π
v
M
Q
.
∼
Thus,
we
have
a
natural
isomorphism
Q
→
Q
v
.
Next,
let
e
be
a
node
of
E
1
;
b,
b
the
two
distinct
branches
of
e;
v,
v
the
[not
necessarily
distinct]
vertices
of
E
1
to
which
b,
b
abut.
Then
since
the
quotient
Q
[≃
Q
v
≃
Q
v
]
is
finite,
one
may
verify
easily
that
—
after
possibly
replacing
G
1
→
G
by
a
connected
finite
étale
Galois
covering
of
G
that
factors
through
G
1
→
G
—
the
kernels
of
the
two
composites
Π
(E
1
)
e
→
Π
(E
1
)
v
Q
v
,
Π
(E
1
)
e
→
Π
(E
1
)
v
Q
v
—
where
Π
(E
1
)
e
→
Π
(E
1
)
v
,
Π
(E
1
)
e
→
Π
(E
1
)
v
are
the
natural
outer
injections
corresponding
to
b,
b
,
respectively
—
coincide.
Moreover,
if
we
write
N
e
⊆
Π
(E
1
)
e
for
this
kernel,
then
it
follows
immediately
from
condition
(b)
of
Definition
3.4,
(i),
that
the
∼
∼
actions
of
Q
induced
by
the
natural
isomorphisms
Q
→
Q
v
←
Π
(E
1
)
e
/N
e
,
∼
∼
Q
→
Q
v
←
Π
(E
1
)
e
/N
e
on
the
connected
finite
étale
Galois
covering
of
(E
1
)
e
corresponding
to
N
e
⊆
Π
(E
1
)
e
coincide.
Therefore,
since
the
underlying
semi-graph
of
E
1
is
finite,
by
applying
this
argument
to
the
various
nodes
of
E
1
and
then
gluing
the
connected
finite
étale
Galois
coverings
of
the
various
(E
1
)
v
’s
corresponding
to
the
quotients
Π
(E
1
)
v
Q
v
to
one
another
by
means
of
Q-equivariant
isomorphisms,
we
obtain
a
connected
finite
étale
Galois
covering
E
2
→
E
1
which
satisfies
the
condition
(†
1
Q,G
1
).
Write
E
2
0
→
E
for
the
Galois
closure
of
the
connected
finite
étale
covering
E
2
→
E;
thus,
since
E
1
is
Galois
over
E,
we
have
connected
finite
étale
Galois
coverings
E
2
0
→
E
2
→
E
1
of
E
1
.
Now
it
follows
Combinatorial
anabelian
topics
I
59
immediately
from
the
condition
(†
1
Q,G
1
)
that
E
2
→
E
1
induces
an
iso-
morphism
between
the
underlying
semi-graphs.
In
particular,
it
fol-
lows
from
Lemma
3.6
below,
in
light
of
the
claim
(∗
1
),
that
the
natural
outer
homomorphisms
Π
E
2
→
Π
E
1
Π
G
1
induce
outer
isomorphisms
∼
∼
Π
E
2
/Π
vert
→
Π
E
1
/Π
vert
→
Π
G
1
/Π
vert
≃
π
1
top
(G
1
)
Σ
,
where
we
write
E
2
E
1
G
1
vert
“Π
(−)
⊆
Π
(−)
”
for
the
normal
closed
subgroup
of
“Π
(−)
”
topologically
normally
generated
by
the
verticial
subgroups
and
π
1
top
(G
1
)
Σ
for
the
pro-Σ
completion
of
the
[discrete]
topological
fundamental
group
of
the
underlying
semi-graph
G
1
of
G
1
.
On
the
other
hand,
since
for
each
ver-
∼
∼
tex
v
∈
Vert(E)
→
Vert(G)
and
each
vertex
w
∈
Vert(E
1
)
→
Vert(G
1
)
∼
that
lies
over
v,
the
isomorphism
t
w
:
M
×
Π
w
→
Π
(E
1
)
w
arises
from
the
∼
isomorphism
t
v
:
M
×
Π
v
→
Π
E
v
,
one
may
verify
easily
that
the
closed
subgroup
Π
(E
2
)
w
⊆
Π
E
v
is
normal.
[Here,
we
regard
w
∈
Vert(E
1
)
as
∼
an
element
of
Vert(E
2
)
by
the
bijection
Vert(E
2
)
→
Vert(E
1
)
induced
by
E
2
→
E
1
.]
In
particular,
it
follows
immediately
that
the
connected
finite
étale
Galois
covering
E
2
0
→
E
2
arises
from
a
normal
open
sub-
∼
group
of
the
quotient
Π
E
2
Π
E
2
/Π
vert
→
π
1
(G
1
)
Σ
.
Therefore,
there
E
2
exists
a
connected
finite
étale
Galois
covering
G
1
→
G
that
factors
through
G
1
→
G
[and
arises
from
a
normal
open
subgroup
of
the
quo-
tient
Π
G
1
π
1
top
(G
1
)
Σ
]
such
that
the
connected
finite
étale
covering
E
2
×
G
1
G
1
of
E
is
Galois.
Now
it
follows
immediately
from
the
fact
that
E
2
→
E
satisfies
the
condition
(†
1
Q,G
1
)
that
E
2
×
G
1
G
1
→
E
satisfies
both
conditions
(†
1
Q,G
1
)
and
(†
2
Q,G
1
),
as
desired.
This
completes
the
proof
of
the
claim
(∗
2
).
Next,
we
claim
that
∼
(∗
3
):
the
composite
E
→
E/α
→
G,
together
with
the
composites
M
→
Π
E
v
→
Π
E
for
v
∈
Vert(E),
determine
an
exact
sequence
of
profi-
nite
groups
1
−→
M
−→
Π
E
−→
Π
G
−→
1
.
Indeed,
it
follows
immediately
from
the
claim
(∗
2
)
—
by
arguing
as
in
the
final
portion
of
the
proof
of
(∗
2
)
—
that
any
connected
finite
étale
Galois
covering
of
E
is
a
subcovering
of
a
covering
of
E
which
satisfies
the
condition
(†
Q,G
1
)
for
some
finite
quotient
M
Q
of
M
and
some
connected
finite
étale
Galois
covering
G
1
of
G.
Therefore,
the
exactness
of
the
sequence
in
question
follows
immediately
from
the
various
definitions
involved,
together
with
the
claim
(∗
1
).
This
completes
the
proof
of
the
claim
(∗
3
).
60
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Finally,
we
claim
that
(∗
4
):
the
exact
sequence
of
profinite
groups
1
−→
M
−→
Π
E
−→
Π
G
−→
1
of
(∗
3
)
is
central,
i.e.,
if
we
write
ρ
:
Π
G
→
Aut(M
)
for
the
representation
of
Π
G
on
M
determined
by
this
extension
Π
E
,
then
ρ
is
trivial.
Indeed,
it
follows
immediately
from
condition
(a)
of
Definition
3.4,
(i),
⊆
Ker(ρ),
where
we
write
Π
vert
⊆
Π
G
for
the
normal
closed
that
Π
vert
G
G
subgroup
of
Π
G
topologically
normally
generated
by
the
verticial
sub-
groups
of
Π
G
.
On
the
other
hand,
it
follows
immediately
from
con-
dition
(b)
of
Definition
3.4,
(i),
by
“parallel
transporting”
along
loops
on
G,
that
the
restriction
to
π
1
top
(G)
⊆
π
1
top
(G)
Σ
of
the
representa-
∼
→]
π
1
top
(G)
Σ
→
Aut(M
)
[cf.
Lemma
3.6
below]
tion
[Π
G
Π
G
/Π
vert
G
induced
by
ρ
—
where
we
write
π
1
top
(G)
for
the
[discrete]
topological
fundamental
group
of
the
semi-graph
G
and
π
1
top
(G)
Σ
for
the
pro-Σ
completion
of
π
1
top
(G)
—
is
trivial.
In
particular,
since
the
subgroup
π
1
top
(G)
⊆
π
1
top
(G)
Σ
is
dense,
the
representation
ρ
is
trivial,
as
desired.
This
completes
the
proof
of
the
claim
(∗
4
),
hence
also
the
proof
of
as-
sertion
(i).
Assertion
(ii)
follows
immediately
from
the
various
definitions
in-
volved.
Next,
we
verify
assertion
(iii).
It
follows
immediately
from
assertion
(i),
together
with
Definition
3.4,
(iii),
that
if
we
write
Π
G
E
for
the
pro-Σ
fundamental
group
of
G
E
,
then
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
M
−→
Π
G
E
−→
Π
G
−→
1
.
On
the
other
hand,
it
follows
immediately
from
the
definition
of
G
E
that
one
may
construct
a
tautological
profinite
covering
of
G
E
[i.e.,
a
pro-object
of
the
category
B(G
E
)
that
appears
in
the
discussion
fol-
lowing
[SemiAn],
Definition
2.1]
equipped
with
a
tautological
action
by
E.
In
particular,
one
obtains
an
outer
surjection
Π
G
E
E
that
is
compatible
with
the
respective
outer
surjections
to
Π
G
.
Thus,
one
con-
cludes
from
the
“Five
Lemma”
that
this
outer
surjection
Π
G
E
E
is
an
outer
isomorphism,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Assertion
(iv)
follows
immediately,
in
light
of
assertions
(i),
(ii),
(iii),
from
the
various
definitions
involved.
This
completes
the
proof
of
Lemma
3.5.
Q.E.D.
Combinatorial
anabelian
topics
I
61
Lemma
3.6
(Quotients
by
verticial
subgroups).
Let
H
be
a
semi-graph
of
anabelioids.
Write
Π
H
for
the
pro-Σ
fundamental
group
of
H
[i.e.,
the
pro-Σ
quotient
of
the
fundamental
group
of
H]
and
Π
vert
H
⊆
Π
H
for
the
normal
closed
subgroup
of
Π
H
topologically
normally
gen-
erated
by
the
verticial
subgroups
of
Π
H
.
Then
the
natural
injection
→
Π
H
determines
an
exact
sequence
of
profinite
groups
Π
vert
H
top
Σ
1
−→
Π
vert
H
−→
Π
H
−→
π
1
(H)
−→
1
—
where
we
write
π
1
top
(H)
Σ
for
the
pro-Σ
completion
of
the
[discrete]
topological
fundamental
group
π
1
top
(H)
of
the
underlying
semi-graph
H
of
H.
Proof.
This
follows
immediately
from
the
various
definitions
in-
volved.
Q.E.D.
Theorem
3.7
(Properties
of
the
second
cohomology
group
with
compact
supports).
Let
Σ
be
a
nonempty
set
of
prime
num-
bers,
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
and
M
a
finitely
Σ
-module.
Then
the
following
hold:
generated
Z
(i)
(Change
of
coefficients)
There
exists
a
natural
isomor-
Σ
-modules
phism
of
Z
∼
Σ
)
⊗
Σ
M
H
c
2
(G,
M
)
−→
H
c
2
(G,
Z
Z
that
is
functorial
with
respect
to
isomorphisms
of
the
pair
(G,
M
).
If,
moreover,
Cusp(G)
=
∅,
then
there
exists
a
natu-
Σ
-modules
ral
isomorphism
of
Z
∼
H
c
2
(G,
M
)
−→
H
2
(G,
M
)
that
is
functorial
with
respect
to
isomorphisms
of
the
pair
(G,
M
).
(ii)
(Structure
as
an
abstract
profinite
group)
The
second
co-
homology
group
with
compact
supports
H
c
2
(G,
M
)
of
G
is
[non-
canonically]
isomorphic
to
M
.
(iii)
(Synchronization
with
respect
to
generization)
Let
S
⊆
Node(G)
be
a
subset
of
Node(G).
Then
the
specialization
∼
outer
isomorphism
Φ
G
S
:
Π
G
S
→
Π
G
with
respect
to
S
[cf.
Definition
2.10]
determines
a
natural
isomorphism
∼
H
c
2
(G,
M
)
−→
H
c
2
(G
S
,
M
)
62
Yuichiro
Hoshi
and
Shinichi
Mochizuki
that
is
functorial
with
respect
to
isomorphisms
of
the
triple
(G,
S,
M
).
(iv)
(Synchronization
with
respect
to
“surgery”)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G,
S
⊆
Node(G|
H
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
separating
type
[cf.
Definition
2.5,
(i)],
and
T
⊆
Cusp((G|
H
)
S
)
[cf.
Definition
2.5,
(ii)]
an
omittable
[cf.
Definition
2.4,
(i)]
subset
of
Cusp((G|
H
)
S
).
Then
there
exists
a
natural
isomorphism
—
given
by
“extension
by
zero”
—
∼
H
c
2
(((G|
H
)
S
)
•T
,
M
)
−→
H
c
2
(G,
M
)
[cf.
Definition
2.4,
(ii)]
that
is
functorial
with
respect
to
iso-
morphisms
of
the
quintuple
(G,
H,
S,
T,
M
).
In
particular,
for
each
vertex
v
∈
Vert(G)
of
G,
there
exists
a
natural
isomor-
Σ
-modules
phism
of
Z
∼
H
c
2
(v,
M
)
−→
H
c
2
(G,
M
)
[cf.
Remark
2.5.1,
(ii)]
that
is
functorial
with
respect
to
iso-
morphisms
of
the
triple
(G,
v,
M
).
(v)
(Homomorphisms
induced
by
finite
étale
coverings)
Let
H
→
G
be
a
connected
finite
étale
covering
of
G.
Then
the
image
of
the
natural
homomorphism
H
c
2
(G,
M
)
−→
H
c
2
(H,
M
)
is
given
by
[Π
G
:
Π
H
]
·
H
c
2
(H,
M
)
.
Proof.
Assertion
(iii)
follows
immediately
from
condition
(1)
of
Proposition
2.9,
(i).
Next,
we
verify
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅.
∼
Σ
)
⊗
Σ
M
The
existence
of
a
natural
isomorphism
H
c
2
(G,
M
)
→
H
c
2
(G,
Z
Z
follows
immediately
from
Lemma
3.3.
On
the
other
hand,
the
fact
that
H
c
2
(G,
M
)
is
[noncanonically]
isomorphic
to
M
follows
immediately
from
Lemma
3.3,
together
with
the
following
well-known
facts
[cf.
[CmbGC],
Remark
1.1.3]:
(A)
Π
G
is
a
free
pro-Σ
group.
Combinatorial
anabelian
topics
I
63
(B)
For
any
cusp
e
0
∈
Cusp(G)
of
G,
the
natural
homomorphism
Σ
-modules
of
Z
Π
e
−→
Π
ab
G
e∈Cusp(G)\{e
0
}
Σ
-modules
[cf.
the
discussion
entitled
is
a
split
injection
of
free
Z
“Topological
groups”
in
§0],
and
its
image
contains
the
image
of
Π
e
0
in
Π
ab
G
.
This
completes
the
proof
of
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅.
Next,
we
verify
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅.
∼
The
existence
of
a
natural
isomorphism
H
c
2
(G,
M
)
→
H
2
(G,
M
)
is
well-
known
[cf.,
e.g.,
[NSW],
Theorem
2.7.7].
Now
it
follows
from
assertion
(iii)
that
to
verify
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅,
we
may
assume
without
loss
of
generality
—
by
replacing
G
by
G
Node(G)
—
that
Node(G)
=
∅.
Then
the
existence
of
a
natural
isomorphism
∼
Σ
)
⊗
Σ
M
and
the
fact
that
H
2
(G,
M
)
is
[non-
H
c
2
(G,
M
)
→
H
c
2
(G,
Z
c
Z
canonically]
isomorphic
to
M
follow
immediately
from
the
existence
of
∼
a
natural
isomorphism
H
c
2
(G,
M
)
→
H
2
(G,
M
)
and
the
fact
that
any
compact
Riemann
surface
of
genus
=
0
is
a
“K(π,
1)”
space
[i.e.,
its
uni-
versal
covering
is
contractible],
together
with
the
well-known
structure
of
the
second
cohomology
group
of
a
compact
Riemann
surface.
This
completes
the
proof
of
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅.
Next,
we
verify
assertion
(iv)
in
the
case
where
H
=
G
and
S
=
∅,
i.e.,
((G|
H
)
S
)
•T
=
G
•T
.
Thus,
suppose
that
H
=
G
and
S
=
∅.
Now
Σ
-modules
define
a
homomorphism
of
Z
H
c
2
(G
•T
,
M
)
−→
H
c
2
(G,
M
)
as
follows:
Let
G
•T
→
G
•T
be
a
universal
covering
of
G
•T
which
is
compatible
[in
the
evident
sense]
with
the
universal
covering
G
→
G
of
G,
s
•
a
section
of
the
natural
surjection
Cusp(
G
•T
)
Cusp(G
•T
),
and
[E
•
,
(ι
•
e
:
Π
s
•
(e)
→
E
•
)
e∈Cusp(G
•T
)
]
∈
H
c
2
(G
•T
,
M
)
an
element
of
H
c
2
(G
•T
,
M
).
Write
E
for
the
fiber
product
of
the
surjection
E
•
Π
G
•T
and
the
natural
surjection
Π
G
Π
G
•T
[arising
from
the
compatibility
of
the
respective
universal
coverings].
Next,
we
introduce
notation
as
follows:
•
for
e
∈
Cusp(G
•T
)
(=
Cusp(G)
\
T
⊆
Cusp(G)),
denote
by
ι
e
:
Π
e
→
E
—
where
we
use
the
notation
Π
e
⊆
Π
G
to
de-
note
an
edge-like
subgroup
of
Π
G
associated
to
e
such
that
the
64
Yuichiro
Hoshi
and
Shinichi
Mochizuki
composite
Π
e
→
Π
G
Π
G
•T
determines
an
isomorphism
of
Π
e
with
Π
s
•
(e)
⊆
Π
G
•T
—
the
section
over
Π
e
naturally
deter-
mined
by
the
composite
ι
•
∼
e
E
•
,
Π
e
−→
Π
s
•
(e)
−→
and
•
for
e
∈
Cusp(G)
\
Cusp(G
•T
)
(=
T
⊆
Cusp(G)),
denote
by
ι
e
:
Π
e
→
E
—
where
we
use
the
notation
Π
e
⊆
Π
G
to
denote
an
edge-like
subgroup
of
Π
G
associated
to
e
—
the
section
over
Π
e
naturally
determined
by
the
trivial
homomorphism
Π
e
→
E
•
.
Then
it
follows
immediately
from
the
various
definitions
involved
that
the
assignment
“[E
•
,
(ι
•
e
)
e∈Cusp(G
•T
)
]
→
(E,
(ι
e
)
e∈Cusp(G)
)”
determines
a
Σ
-modules
homomorphism
of
Z
H
c
2
(G
•T
,
M
)
−→
H
c
2
(G,
M
)
,
as
desired.
Next,
we
verify
that
this
homomorphism
H
c
2
(G
•T
,
M
)
→
H
c
2
(G,
M
)
is
an
isomorphism.
First,
let
us
observe
that
it
follows
from
assertion
(ii)
that,
to
verify
that
the
homomorphism
in
question
is
an
isomorphism,
it
suffices
to
verify
that
it
is
surjective.
The
rest
of
the
proof
of
assertion
(iv)
in
the
case
where
H
=
G
and
S
=
∅
is
devoted
to
verifying
this
surjectivity.
To
verify
the
desired
surjectivity,
by
induction
on
the
car-
dinality
T
of
the
finite
set
T
,
we
may
assume
without
loss
of
generality
that
T
=
1,
i.e.,
T
=
{e
0
}
for
some
e
0
∈
Cusp(G).
To
verify
the
desired
surjectivity,
let
[E,
(ι
e
)
e∈Cusp(G)
]
∈
H
c
2
(G,
M
)
be
an
element
of
H
c
2
(G,
M
).
Then
since
Π
G
is
a
free
pro-Σ
group,
there
exists
a
continuous
section
Π
G
→
E
of
the
surjection
E
Π
G
,
hence
also
∼
—
since
the
extension
E
of
Π
G
is
central
—
an
isomorphism
M
×Π
G
→
E.
Write
Π
G
Π
for
the
maximal
cuspidally
central
quotient
[cf.
[AbsCsp],
Definition
1.1,
(i)]
relative
to
the
surjection
Π
G
Π
G
•T
,
E
Π
for
the
quotient
of
E
by
the
normal
closed
subgroup
of
E
corresponding
to
∼
{1}
×
Ker(Π
G
Π)
⊆
M
×
Π
G
[thus,
E
Π
←
M
×
Π],
and
N
⊆
E
Π
for
the
image
of
the
composite
ι
e
0
Π
s(e
0
)
→
E
E
Π
.
Now
we
claim
that
N
⊆
E
Π
is
contained
in
the
center
Z(E
Π
)
of
E
Π
,
hence
also
normal
in
E
Π
.
Indeed,
since
the
composite
Π
s(e
0
)
→
Π
G
Π
Combinatorial
anabelian
topics
I
65
is
injective,
and
its
image
coincides
with
the
kernel
of
the
natural
sur-
jection
Π
Π
G
•T
,
it
holds
that
the
image
of
the
composite
ι
e
0
∼
Π
s(e
0
)
→
E
E
Π
←
M
×
Π
is
contained
in
M
×
Ker(Π
→
Π
G
•T
).
On
the
other
hand,
since
the
extension
E
of
Π
G
is
central,
it
follows
from
the
definition
of
the
quotient
∼
Π
of
Π
G
that
the
image
of
M
×
Ker(Π
Π
G
•T
)
in
E
Π
via
M
×
Π
→
E
Π
is
contained
in
the
center
Z(E
Π
)
of
E
Π
.
This
completes
the
proof
of
the
above
claim.
Now
it
follows
from
the
definition
of
N
⊆
E
Π
,
together
with
the
above
claim,
that
we
obtain
a
commutative
diagram
of
profinite
groups
1
−−−−→
M
−−−−→
E
⏐
⏐
−−−−→
Π
G
⏐
⏐
−−−−→
1
1
−−−−→
M
−−−−→
E
Π
/N
−−−−→
Π
G
•T
−−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
surjective.
In
particular,
we
obtain
an
extension
E
Π
/N
of
Π
G
•T
by
M
,
which
is
central
since
the
extension
E
is
central.
For
e
∈
Cusp(G
•T
)
=
Cusp(G)
\
{e
0
},
write
Π
•
e
⊆
Π
G
•T
for
the
edge-like
subgroup
of
Π
G
•T
[associated
to
e
∈
Cusp(G
•T
)]
determined
by
the
image
of
Π
s(e)
⊆
Π
G
and
ι
•
e
for
the
section
Π
•
e
→
E
Π
/N
over
Π
•
e
determined
by
ι
e
:
Π
s(e)
→
E.
Then
it
follows
immediately
from
the
various
definitions
involved
that
the
image
of
[E
Π
/N,
(ι
•
e
)
e
∈Cusp(G
•T
)
]
∈
H
c
2
(G
•T
,
M
)
in
H
c
2
(G,
M
)
is
[E,
(ι
e
)
e∈Cusp(G)
]
∈
H
c
2
(G,
M
).
This
completes
the
proof
of
the
desired
surjectivity
and
hence
of
assertion
(iv)
in
the
case
where
H
=
G
and
S
=
∅.
Next,
to
complete
the
proof
of
assertion
(iv)
in
the
general
case,
one
verifies
immediately
that
it
suffices
to
verify
assertion
(iv)
in
the
case
where
T
=
∅,
i.e.,
((G|
H
)
S
)
•T
=
(G|
H
)
S
.
Thus,
suppose
that
def
Σ
-
T
=
∅.
Write
H
=
(G|
H
)
S
.
To
define
a
natural
homomorphism
of
Z
2
2
→
H
be
a
universal
covering
of
modules
H
c
(H,
M
)
→
H
c
(G,
M
),
let
H
H
which
is
compatible
[in
the
evident
sense]
with
the
universal
covering
Cusp(H),
G
→
G
of
G,
s
H
a
section
of
the
natural
surjection
Cusp(
H)
H
2
:
Π
→
E
)
]
∈
H
(H,
M
)
an
element
of
and
[E
H
,
(ι
H
s
H
(e)
e∈Cusp(H)
e
c
2
H
H
c
(H,
M
).
Since
the
extension
E
of
Π
H
by
M
is
central,
the
sec-
H
naturally
determines
an
isomorphism
tion
ι
H
e
:
Π
s
H
(e)
→
E
∼
M
×
Π
s
H
(e)
−→
E
H
×
Π
H
Π
s
H
(e)
66
Yuichiro
Hoshi
and
Shinichi
Mochizuki
of
the
direct
product
M
×
Π
s
H
(e)
with
the
fiber
product
E
H
×
Π
H
Π
s
H
(e)
of
the
surjection
E
H
Π
H
and
the
natural
inclusion
Π
s
H
(e)
→
Π
H
.
Write
G
E
H
for
the
semi-graph
of
anabelioids
associated
to
the
central
extension
E
H
[cf.
Definition
3.4,
(ii)].
Then
one
may
define
a
central
extension
of
G
by
M
∼
(E,
α,
β
:
E/α
→
G)
[cf.
Definition
3.4,
(i)]
whose
restriction
to
H,
relative
to
the
isomor-
∼
phism
β
:
E/α
→
G,
is
isomorphic
to
the
semi-graph
of
anabelioids
G
E
H
as
follows:
We
take
the
underlying
semi-graph
of
E
to
be
the
underlying
semi-graph
of
G;
for
each
vertex
v
∈
Vert(G|
H
),
we
take
the
anabelioid
E
v
of
E
corresponding
to
the
vertex
v
∈
Vert(G|
H
)
to
be
the
anabelioid
(G
E
H
)
v
of
G
E
H
corresponding
to
the
vertex
v;
for
each
vertex
v
∈
Vert(G)
\
Vert(G|
H
),
we
take
the
anabelioid
E
v
of
E
corresponding
to
v
∈
Vert(G)
\
Vert(G|
H
)
to
be
the
anabelioid
associ-
ated
to
the
profinite
group
M
×
Π
v
.
Then
the
above
isomorphisms
∼
M
×
Π
s
H
(e)
→
E
H
×
Π
H
Π
s
H
(e)
induced
by
the
various
ι
H
e
’s
naturally
determine
the
remaining
data
[i.e.,
consisting
of
anabelioids
associated
to
edges
and
morphisms
of
anabelioids
associated
to
branches]
necessary
to
define
a
semi-graph
of
anabelioids
E
which
is
naturally
equipped
with
a
structure
of
central
extension
of
G
by
M
whose
restriction
to
H
is
naturally
isomorphic
to
the
semi-graph
of
anabelioids
G
E
H
,
as
desired.
Now
it
follows
from
Lemma
3.5,
(i),
that
if
we
denote
by
Π
E
the
pro-Σ
fundamental
group
of
E
—
i.e.,
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
E
—
then
Π
E
is
a
central
extension
of
Π
G
by
M
.
Thus,
it
follows
from
the
equivalences
of
categories
of
Lemma
3.5,
(iv),
that
the
sections
ι
H
e
—
where
e
ranges
over
the
cusps
of
G
that
abut
to
a
vertex
of
G|
H
—
and
the
tautological
sections
Π
e
→
M
×
Π
e
=
Π
E
e
—
where
e
ranges
over
the
cusps
of
G
that
do
not
abut
to
a
vertex
of
G|
H
—
naturally
determine
an
equivalence
class
[Π
E
,
(ι
e
)
e∈Cusp(G)
]
∈
H
c
2
(G,
M
).
In
particular,
we
obtain
a
map
H
c
2
(H,
M
)
−→
H
c
2
(G,
M
)
by
assigning
[E
H
,
(ι
H
e
)
e∈Cusp(H)
]
→
[Π
E
,
(ι
e
)
e∈Cusp(G)
].
Moreover,
it
fol-
lows
immediately
from
the
various
definitions
involved
that
this
map
is
Σ
-modules,
as
desired.
a
homomorphism
of
Z
Next,
we
verify
that
this
homomorphism
H
c
2
(H,
M
)
→
H
c
2
(G,
M
)
is
an
isomorphism.
Since,
for
any
vertex
v
∈
Vert(G|
H
),
the
natural
morphism
G|
v
→
G
factors
through
(G|
H
)
S
=
H
→
G,
by
replacing
H
by
G|
v
[cf.
Remark
2.5.1,
(ii)],
we
may
assume
without
loss
of
generality
that
H
=
G|
v
.
Moreover,
if
Node(G)
=
∅,
then
assertion
(iv)
in
the
case
where
T
=
∅
is
immediate;
thus,
we
may
assume
without
loss
Combinatorial
anabelian
topics
I
67
of
generality
that
Node(G)
=
∅.
On
the
other
hand,
it
follows
from
assertion
(ii)
that
to
verify
that
the
homomorphism
in
question
is
an
isomorphism,
it
suffices
to
verify
that
it
is
surjective.
The
rest
of
the
proof
of
assertion
(iv)
in
the
case
where
T
=
∅
is
devoted
to
verifying
the
surjectivity
of
the
homomorphism
H
c
2
(v,
M
)
→
H
c
2
(G,
M
).
Let
J
be
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
such
that
there
exist
a
vertex
w
∈
Vert(J
)
and
an
“omittable”
cusp
e
∈
C(w)
[i.e.,
a
cusp
that
abuts
to
w
such
that
{e}
is
omittable]
such
that
J
•{e}
∼
is
isomorphic
to
G,
and,
moreover,
the
isomorphism
J
•{e}
→
G
induces
∼
an
isomorphism
of
(J
|
w
)
•{e}
→
G|
v
.
[Note
that
one
may
verify
easily
that
such
a
semi-graph
of
anabelioids
of
pro-Σ
PSC
type
always
exists.]
Then
it
follows
immediately
from
assertion
(iv)
in
the
case
where
H
=
G
and
S
=
∅,
together
with
the
various
definitions
involved,
that
we
have
a
commutative
diagram
∼
∼
H
c
2
(v,
M
)
−−−−→
H
c
2
((J
|
w
)
•{e}
,
M
)
−−−−→
H
c
2
(w,
M
)
⏐
⏐
⏐
⏐
∼
H
c
2
(G,
M
)
−−−−→
H
c
2
(J
•{e}
,
M
)
∼
−−−−→
H
c
2
(J
,
M
)
—
where
the
left-hand
horizontal
arrows
are
isomorphisms
induced
by
∼
∼
the
isomorphisms
(J
|
w
)
•{e}
→
G|
v
,
J
•{e}
→
G,
respectively,
and
the
right-hand
horizontal
arrows
are
isomorphisms
obtained
by
applying
as-
sertion
(iv)
in
the
case
where
H
=
G
and
S
=
∅.
In
particular,
to
verify
the
desired
surjectivity
of
the
homomorphism
H
c
2
(v,
M
)
→
H
c
2
(G,
M
),
by
replacing
G
(respectively,
v)
by
J
(respectively,
w),
we
may
assume
without
loss
of
generality
that
C(v)
=
∅.
To
verify
the
desired
surjectivity
of
the
homomorphism
H
c
2
(v,
M
)
→
2
H
c
(G,
M
)
in
the
case
where
C(v)
=
∅,
let
[E,
(ι
e
)
e∈Cusp(G)
]
∈
H
c
2
(G,
M
)
be
an
element
of
H
c
2
(G,
M
).
Now
it
follows
from
Lemma
3.3,
together
with
the
assumption
that
C(v)
=
∅,
that
we
have
two
exact
sequences
Σ
-modules
of
Z
Hom
Z
Σ
(Π
ab
Hom
Z
Σ
(Π
e
,
M
)
−→
H
c
2
(G,
M
)
−→
0
;
G
,
M
)
−→
e∈Cusp(G)
Hom
Z
Σ
(Π
ab
G|
v
,
M
)
−→
Hom
Z
Σ
(Π
e
,
M
)
−→
H
c
2
(v,
M
)
−→
0
.
e∈Cusp(G|
v
)
Let
e
0
∈
C(v)
be
a
cusp
of
G
that
abuts
to
v.
Here,
note
that
it
fol-
lows
immediately
from
the
definition
of
G|
v
that
e
0
may
be
regarded
as
a
cusp
of
G|
v
.
Then
it
follows
immediately
from
the
facts
(A),
(B)
used
in
the
proof
of
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅
68
Yuichiro
Hoshi
and
Shinichi
Mochizuki
that
there
exists
a
lifting
(φ
e
)
e∈Cusp(G)
∈
Σ
(Π
e
,
M
)
e∈Cusp(G)
Hom
Z
of
[E,
(ι
e
)
e∈Cusp(G)
]
∈
H
c
2
(G,
M
)
[with
respect
to
the
first
exact
se-
quence
of
the
above
display]
such
that
if
e
=
e
0
,
then
φ
e
=
0.
Write
(ψ
e
)
e∈Cusp(G|
v
)
∈
e∈Cusp(G|
v
)
Hom
Z
Σ
(Π
e
,
M
)
for
the
element
such
that
ψ
e
0
=
φ
e
0
,
ψ
e
=
0
for
e
=
e
0
.
Then
it
follows
immediately
from
the
definitions
of
the
above
exact
sequences
and
the
homomorphism
H
2
(v,
M
)
→
H
c
2
(G,
M
)
in
question
that
the
image
of
(ψ
e
)
e∈Cusp(G|
v
)
∈
c
2
Σ
(Π
e
,
M
)
in
H
c
(v,
M
)
is
mapped
to
[E,
(ι
e
)
e∈Cusp(G)
]
e∈Cusp(G|
v
)
Hom
Z
2
∈
H
c
(G,
M
)
via
the
homomorphism
H
c
2
(v,
M
)
→
H
c
2
(G,
M
)
in
question.
This
completes
the
proof
of
assertion
(iv)
in
the
case
where
T
=
∅,
hence
also
of
assertion
(iv)
in
the
general
case.
Finally,
we
verify
assertion
(v).
If
Cusp(G)
=
∅,
then
it
follows
im-
mediately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
assertions
(i),
(ii)
in
the
case
where
Cusp(G)
=
∅,
together
with
the
well-known
structure
of
the
second
cohomology
group
of
a
compact
Rie-
mann
surface,
that
assertion
(v)
holds.
Next,
suppose
that
Cusp(G)
=
∅.
Write
G
for
the
double
of
G
[cf.
[CmbGC],
Proposition
2.2,
(i)]
—
i.e.,
the
analogue
in
the
theory
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-
type
to
the
well-known
“double”
of
a
Riemann
surface
with
boundary.
Write
H
for
the
double
of
H.
Then
it
follows
from
the
various
defini-
tions
involved
that
the
connected
finite
étale
covering
H
→
G
determines
a
connected
finite
étale
covering
H
→
G
of
degree
[Π
G
:
Π
H
].
Next,
let
us
observe
G
(respectively,
H)
may
be
naturally
identified
with
the
restriction
[cf.
Definition
2.2,
(ii)]
of
G
(respectively,
H
)
to
a
suit-
able
sub-semi-graph
of
PSC-type
of
the
underlying
semi-graph
of
G
(respectively,
H
).
Thus,
it
follows
from
assertion
(iv)
that
we
have
a
Σ
-modules
commutative
diagram
of
Z
∼
H
c
2
(G,
M
)
−−−−→
H
c
2
(G
,
M
)
⏐
⏐
⏐
⏐
∼
H
c
2
(H,
M
)
−−−−→
H
c
2
(H
,
M
)
—
where
the
horizontal
arrows
are
the
isomorphisms
of
assertion
(iv),
and
the
vertical
arrows
are
the
homomorphisms
induced
by
the
con-
nected
finite
étale
coverings
H
→
G,
H
→
G
,
respectively
—
and
hence
that
assertion
(v)
in
the
case
where
Cusp(G)
=
∅
follows
immedi-
ately
from
assertion
(v)
in
the
case
where
Cusp(G)
=
∅.
This
completes
the
proof
of
assertion
(v).
Q.E.D.
Combinatorial
anabelian
topics
I
69
Definition
3.8.
(i)
We
shall
write
Σ
),
Z
Σ
)
Λ
G
=
Hom
Z
Σ
(H
c
2
(G,
Z
def
and
refer
to
Λ
G
as
the
cyclotome
associated
to
G.
For
a
vertex
v
∈
Vert(G)
of
G,
we
shall
write
Σ
),
Z
Σ
)
Λ
v
=
Hom
Z
Σ
(H
c
2
(v,
Z
def
and
refer
to
Λ
v
as
the
cyclotome
associated
to
v
∈
Vert(G).
Note
that
it
follows
from
Theorem
3.7,
(ii),
that
the
cyclotomes
Σ
-modules
of
rank
1.
Λ
G
and
Λ
v
are
free
Z
(ii)
We
shall
write
Σ
)
∗
χ
G
:
Aut(G)
−→
Aut(Λ
G
)
≃
(
Z
for
the
natural
homomorphism
induced
by
the
natural
action
of
Σ
)
and
refer
to
χ
G
as
the
pro-Σ
cyclotomic
Aut(G)
on
H
c
2
(G,
Z
character
of
G.
For
a
vertex
v
∈
Vert(G)
of
G,
we
shall
write
def
Σ
)
∗
χ
v
=
χ
G|
v
:
Aut(G|
v
)
−→
Aut(Λ
v
)
≃
(
Z
and
refer
to
χ
v
as
the
pro-Σ
cyclotomic
character
of
v.
Remark
3.8.1.
One
verifies
easily
that
if
l
∈
Σ,
then
the
composite
χ
G
Σ
)
∗
Z
∗
Aut(G)
→
(
Z
l
coincides
with
the
pro-l
cyclotomic
character
of
Aut(G)
defined
in
the
statement
of
[CmbGC],
Lemma
2.1.
Corollary
3.9
(Synchronization
of
cyclotomes).
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Then
the
following
hold:
(i)
(Synchronization
with
respect
to
generization)
Let
S
⊆
Node(G)
be
a
subset
of
Node(G).
Then
the
specialization
outer
70
Yuichiro
Hoshi
and
Shinichi
Mochizuki
∼
isomorphism
Φ
G
S
:
Π
G
S
→
Π
G
with
respect
to
S
[cf.
Defini-
tion
2.10]
determines
a
natural
isomorphism
∼
Λ
G
S
−→
Λ
G
that
is
functorial
with
respect
to
isomorphisms
of
the
pair
(G,
S).
(ii)
(Synchronization
with
respect
to
“surgery”)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G,
S
⊆
Node(G|
H
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
separating
type
[cf.
Definition
2.5,
(i)],
and
T
⊆
Cusp((G|
H
)
S
)
[cf.
Definition
2.5,
(ii)]
an
omittable
[cf.
Definition
2.4,
(i)]
subset
of
Cusp((G|
H
)
S
).
Then
there
exists
a
natural
isomorphism
—
given
by
“extension
by
zero”
—
∼
Λ
G
−→
Λ
((G|
H
)
S
)
•T
[cf.
Definition
2.4,
(ii)]
that
is
functorial
with
respect
to
iso-
morphisms
of
the
quadruple
(G,
H,
S,
T
).
In
particular,
[by
tak-
ing
the
inverse
of
this
isomorphism]
we
obtain,
for
each
vertex
Σ
-modules
v
∈
Vert(G)
of
G,
a
natural
isomorphism
of
Z
∼
syn
v
:
Λ
v
−→
Λ
G
that
is
functorial
with
respect
to
isomorphisms
of
the
pair
(G,
v).
(iii)
(Synchronization
with
respect
to
finite
étale
coverings)
Let
H
→
G
be
a
connected
finite
étale
covering
of
G.
Then
there
exists
a
natural
isomorphism
∼
Λ
H
−→
Λ
G
that
is
functorial
with
respect
to
isomorphisms
of
the
pair
(G,
H).
(iv)
(Synchronization
of
cyclotomic
characters)
Let
v
∈
Vert(G)
be
a
vertex
of
G
and
α
∈
Aut
{v}
(G)
[cf.
Definition
2.6,
(i)].
Then
it
holds
that
χ
G
(α)
=
χ
v
(α
G|
v
)
[cf.
Definitions
2.14,
(ii);
3.8,
(ii);
Remark
2.5.1,
(ii)].
Combinatorial
anabelian
topics
I
(v)
71
(Synchronization
associated
to
branches)
Let
e
∈
Edge(G)
be
an
edge
of
G,
b
a
branch
of
e
that
abuts
to
a
vertex
v
∈
V(e),
and
Π
e
⊆
Π
G
an
edge-like
subgroup
of
Π
G
associated
to
e
∈
Edge(G).
Then
there
exists
a
natural
isomorphism
∼
syn
b
:
Π
e
−→
Λ
v
that
is
functorial
with
respect
to
isomorphisms
of
the
quadru-
ple
(G,
b,
e,
v).
(vi)
(Difference
between
two
synchronizations
associated
to
the
two
branches
of
a
node)
Let
e
∈
Node(G)
be
a
node
of
G
with
branches
b
1
=
b
2
that
abut
to
vertices
v
1
,
v
2
∈
Vert(G),
respectively.
Then
the
two
composites
syn
b
1
∼
syn
b
2
∼
syn
v
1
∼
syn
v
2
∼
Π
e
−→
Λ
v
1
−→
Λ
G
;
Π
e
−→
Λ
v
2
−→
Λ
G
differ
by
the
automorphism
of
Λ
G
given
by
multiplication
by
Σ
.
−1
∈
Z
Proof.
Assertion
(i)
(respectively,
(ii))
follows
immediately
from
Theorem
3.7,
(iii)
(respectively,
Theorem
3.7,
(iv)).
Assertion
(iv)
fol-
lows
immediately
from
assertion
(ii).
Next,
we
verify
assertion
(iii).
It
follows
immediately
from
Theo-
Σ
-modules
Λ
H
→
Λ
G
obtained
rem
3.7,
(v),
that
the
homomorphism
of
Z
Σ
)”
to
the
induced
homomorphism
by
applying
the
functor
“Hom
Z
Σ
(−,
Z
2
Σ
2
Σ
)
→
H
(H,
Z
)
and
dividing
by
the
index
[Π
G
:
Π
H
]
is
an
iso-
H
c
(G,
Z
c
morphism.
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(v).
First,
we
observe
that
to
verify
as-
sertion
(v),
by
replacing
G
by
G|
v
and
e
∈
Edge(G)
by
the
cusp
of
G|
v
corresponding
to
b,
we
may
assume
without
loss
of
generality
that
Σ
-modules
e
∈
Cusp(G).
Then
we
have
homomorphisms
of
Z
Hom
Z
Σ
(Π
e
,
Π
e
)
→
Σ
(Π
e
,
Π
e
)
e
∈Cusp(G)
Hom
Z
H
c
2
(G,
Π
e
)
∼
→
Hom
Z
Σ
(Λ
G
,
Π
e
)
—
where
the
first
arrow
is
the
natural
inclusion
into
the
component
indexed
by
e,
and
the
second
arrow
is
the
surjection
appearing
in
the
exact
sequence
of
Lemma
3.3
in
the
case
where
M
=
Π
e
.
Here,
we
note
that
it
follows
immediately
from
the
facts
(A),
(B)
used
in
the
proof
of
Theorem
3.7,
(i),
(ii),
that
the
composite
of
these
homomorphisms
is
an
72
Yuichiro
Hoshi
and
Shinichi
Mochizuki
isomorphism.
Therefore,
we
obtain
a
natural
isomorphism
∼
syn
b
:
Π
e
−→
Λ
G
by
forming
the
inverse
of
the
image
of
the
identity
automorphism
of
Π
e
via
the
composite
of
the
homomorphisms
of
the
above
display.
This
completes
the
proof
of
assertion
(v).
Finally,
we
verify
assertion
(vi).
First,
we
observe
that
one
may
verify
easily
that
there
exist
•
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
H
†
,
•
a
sub-semi-graph
of
PSC-type
K
†
of
the
underlying
semi-graph
of
H
†
,
•
an
omittable
subset
S
†
⊆
Cusp((H
†
)|
K
†
),
and
•
an
isomorphism
∼
((H
†
)|
K
†
)
•S
†
−→
G
such
that
the
node
e
H
†
∈
Node(H
†
)
of
H
†
corresponding,
relative
to
∼
the
isomorphism
((H
†
)|
K
†
)
•S
†
→
G,
to
the
node
e
∈
Node(G)
is
not
of
separating
type.
[Note
that
it
follows
immediately
from
the
various
def-
∼
initions
involved
that
Node(G)
←
Node(((H
†
)|
K
†
)
•S
†
)
may
be
regarded
†
as
a
subset
of
Node(H
).]
Thus,
it
follows
immediately
from
assertions
(i),
(ii)
—
by
replacing
G
(respectively,
e)
by
(H
†
)
Node(H
†
)\{e
H†
}
(re-
spectively,
e
H
†
)
—
that
to
verify
assertion
(vi),
we
may
assume
without
loss
of
generality
that
Node(G)
=
{e},
and
that
e
is
not
of
separating
type.
Next,
we
observe
that
one
may
verify
easily
that
there
exists
a
semi-
graph
of
anabelioids
of
pro-Σ
PSC-type
H
‡
such
that
•
Node(H
‡
)
consists
of
precisely
two
elements
e
H
‡
,
e
H
‡
;
•
V(e
H
‡
)
consists
of
precisely
one
element
v
H
‡
of
type
(0,
3)
[cf.
Definition
2.3,
(iii)].
•
e
H
‡
is
of
separating
type;
•
(H
‡
)
{e
‡
}
is
isomorphic
to
G.
H
Thus,
if
we
write
K
‡
for
the
unique
sub-semi-graph
of
PSC-type
of
the
underlying
semi-graph
of
H
‡
whose
set
of
vertices
=
{v
H
‡
},
then
it
fol-
lows
immediately
from
assertions
(i),
(ii)
—
by
replacing
G
(respectively,
e)
by
H
‡
|
K
‡
(respectively,
e
H
‡
)
—
that
to
verify
assertion
(vi),
we
may
assume
without
loss
of
generality
that
Node(G)
=
{e},
that
e
is
not
of
Combinatorial
anabelian
topics
I
73
separating
type
[so
Vert(G)
consists
of
precisely
one
element],
and
that
G
is
of
type
(1,
1).
Write
v
∈
Vert(G)
for
the
unique
vertex
of
G.
Note
that
it
follows
immediately
from
the
various
assumptions
on
G
that
G|
v
is
of
type
(0,
3).
Write
e
1
,
e
2
∈
Cusp(G|
v
)
for
the
cusps
of
G|
v
corresponding,
respectively,
to
the
two
branches
b
1
,
b
2
of
the
node
e;
write
e
3
∈
Cusp(G|
v
)
for
the
unique
element
of
Cusp(G|
v
)
\
{e
1
,
e
2
}.
Then
since
G|
v
is
of
type
(0,
3),
there
exists
a
graphic
isomorphism
of
G|
v
with
the
semi-graph
of
an-
abelioids
of
pro-Σ
PSC-type
[without
nodes]
determined
by
the
tripod
[cf.
the
discussion
entitled
“Curves”
in
§0]
P
1
k
\
{0,
1,
∞}
over
an
alge-
braically
closed
field
k
of
characteristic
∈
Σ
such
that
the
cusps
e
1
,
e
2
of
G|
v
correspond
to
the
cusps
0,
∞
of
P
1
k
\
{0,
1,
∞},
respectively,
relative
to
the
graphic
isomorphism.
Thus,
by
considering
the
automorphism
of
P
1
k
\
{0,
1,
∞}
over
k
given
by
“t
→
1/t”,
we
obtain
an
automorphism
τ
v
∈
Aut(G|
v
)
of
G|
v
that
maps
e
1
→
e
2
,
e
2
→
e
1
.
Moreover,
since
this
automorphism
of
P
1
k
\
{0,
1,
∞}
induces
an
automorphism
of
the
stable
log
curve
of
type
(1,
1)
obtained
by
identifying
the
cusps
0
and
∞
of
P
1
k
\
{0,
1,
∞},
we
also
obtain
an
automorphism
τ
G
∈
Aut(G)
of
G.
Note
that
it
follows
immediately
from
the
definition
of
τ
v
,
together
with
the
well-known
structure
of
the
étale
fundamental
group
of
the
tripod
P
1
k
\
{0,
1,
∞},
that
the
automorphism
τ
v
induces
the
identity
automorphism
of
the
anabeloid
(G|
v
)
e
3
corresponding
to
e
3
.
Next,
let
us
observe
that
it
follows
immediately
from
the
definition
of
G|
v
,
together
with
the
proof
of
assertion
(v),
that
for
i
=
1,
2,
there
∼
exists
a
natural
isomorphism
Π
e
→
Π
e
i
—
where
we
use
the
notations
Π
e
,
Π
e
i
to
denote
edge-like
subgroups
of
Π
G
,
Π
G|
v
associated
to
e,
e
i
,
respectively
—
such
that
the
composite
∼
syn
b
i
∼
syn
v
∼
Π
e
−→
Π
e
i
−→
Λ
v
[=
Λ
G|
v
]
−→
Λ
G
—
where
we
write
b
i
for
the
[unique]
branch
of
e
i
—
coincides
with
the
composite
in
question
syn
bi
∼
syn
v
∼
Π
e
−→
Λ
v
−→
Λ
G
.
Next,
let
us
observe
that
it
follows
immediately
from
the
functori-
ality
portion
of
assertion
(v)
that
the
automorphisms
τ
v
,
τ
G
induce
a
74
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Σ
-modules
commutative
diagram
of
Z
∼
syn
b
syn
v
∼
syn
b
syn
v
i
Π
e
−−−−→
Π
e
1
−−−−→
Λ
v
[=
Λ
G|
v
]
−−−−→
Λ
G
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
2
Π
e
−−−−→
Π
e
2
−−−−→
Λ
v
[=
Λ
G|
v
]
−−−−→
Λ
G
—
where
the
vertical
arrows
are
the
isomorphisms
induced
by
the
au-
tomorphisms
τ
v
,
τ
G
.
Now
by
considering
the
well-known
local
structure
of
a
stable
log
curve
in
a
neighborhood
of
a
node,
one
may
verify
easily
that
the
left-hand
vertical
arrow
in
the
above
diagram
is
the
automor-
Σ
.
Thus,
to
complete
the
phism
of
Π
e
given
by
multiplication
by
−1
∈
Z
verification
of
assertion
(vi),
it
suffices,
in
light
of
the
commutativity
of
the
above
diagram,
to
verify
that
τ
v
∈
Aut(G|
v
)
induces
the
identity
automorphism
of
Λ
G|
v
=
Λ
v
.
On
the
other
hand,
this
follows
immedi-
ately
from
assertion
(v),
applied
to
the
cusp
e
3
,
together
with
the
fact
that
the
automorphism
τ
v
induces
the
identity
automorphism
of
(G|
v
)
e
3
.
This
completes
the
proof
of
assertion
(vi).
Q.E.D.
§4.
Profinite
Dehn
multi-twists
In
the
present
§,
we
introduce
and
discuss
the
notion
of
a
profinite
Dehn
multi-twist.
Although
our
definition
of
this
notion
[cf.
Defini-
tion
4.4
below]
is
entirely
group-theoretic
in
nature,
our
main
result
concerning
this
notion
[cf.
Theorem
4.8
below]
asserts,
in
effect,
that
this
group-theoretic
notion
coincides
with
the
usual
geometric
notion
of
a
“Dehn
multi-twist”.
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Write
G
for
the
underlying
semi-graph
of
G,
Π
G
for
the
[pro-Σ]
fundamental
group
of
G,
and
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Definition
4.1.
We
shall
say
that
G
is
cyclically
primitive
(respec-
tively,
noncyclically
primitive)
if
Node(G)
=
1,
and
the
unique
node
of
G
is
not
of
separating
type
(respectively,
is
of
separating
type)
[cf.
Definition
2.5,
(i)].
Combinatorial
anabelian
topics
I
75
Remark
4.1.1.
If
G
is
cyclically
primitive
(respectively,
noncycli-
cally
primitive),
then
Vert(G)
=
1
(respectively,
2),
and
the
[discrete]
topological
fundamental
group
π
1
top
(G)
of
the
underlying
semi-graph
G
of
G
is
noncanonically
isomorphic
to
Z
(respectively,
is
trivial).
Lemma
4.2
(Structure
of
the
fundamental
group
of
a
non-
cyclically
primitive
semi-graph
of
anabelioids
of
PSC-type).
Suppose
that
G
is
noncyclically
primitive
[cf.
Definition
4.1].
Let
v,
w
∈
Vert(G)
be
the
two
distinct
vertices
of
G
[cf.
Remark
4.1.1];
elements
of
VCN(
G)
such
that
v
(G)
=
v,
w(G)
e
,
v
,
w
∈
VCN(
G)
=
w,
and,
moreover,
e
∈
N
(
v
)
∩
N
(
w).
Then
the
natural
inclusions
Π
e
,
Π
v
,
Π
w
→
Π
G
determine
an
isomorphism
of
pro-Σ
groups
∼
lim
(Π
v
←
Π
e
→
Π
w
)
−→
Π
G
−→
—
where
the
inductive
limit
is
taken
in
the
category
of
pro-Σ
groups.
Proof.
This
may
be
thought
of
as
a
consequence
of
the
“van
Kam-
pen
Theorem”
in
elementary
algebraic
topology.
At
a
more
combinato-
rial
level,
one
may
reason
as
follows:
It
follows
immediately
from
the
simple
structure
of
the
underlying
semi-graph
G
that
there
is
a
natural
equivalence
of
categories
between
•
the
category
of
finite
sets
with
continuous
Π
G
-action
[and
Π
G
-
equivariant
morphisms]
and
•
the
category
of
finite
sets
with
continuous
actions
of
Π
v
,
Π
w
which
restrict
to
the
same
action
on
Π
e
[and
Π
v
-,
Π
w
-equivariant
morphisms].
The
isomorphism
between
Π
G
and
the
inductive
limit
appearing
in
the
statement
of
Lemma
4.2
now
follows
formally
from
this
equivalence
of
categories.
Q.E.D.
Lemma
4.3
(Infinite
cyclic
tempered
covering
of
a
cyclically
primitive
semi-graph
of
anabelioids
of
PSC-type).
Suppose
that
G
is
cyclically
primitive
[cf.
Definition
4.1].
Denote
by
π
1
temp
(G)
the
tempered
fundamental
group
of
G
[cf.
the
discussion
preceding
[SemiAn],
Proposition
3.6],
by
π
1
top
(G)
[≃
Z
—
cf.
Remark
4.1.1]
the
[discrete]
topological
fundamental
group
of
the
underlying
semi-graph
G
of
G,
and
by
G
∞
→
G
the
connected
tempered
covering
of
G
corresponding
to
the
natural
surjection
π
1
temp
(G)
π
1
top
(G)
[where
we
refer
to
[SemiAn],
§3,
76
Yuichiro
Hoshi
and
Shinichi
Mochizuki
concerning
tempered
coverings
of
a
semi-graph
of
anabelioids].
Then
the
following
hold:
(i)
(Exact
sequence)
The
natural
morphism
G
∞
→
G
induces
an
exact
sequence
1
−→
π
1
temp
(G
∞
)
−→
π
1
temp
(G)
−→
π
1
top
(G)
−→
1
.
Moreover,
the
subgroup
π
1
temp
(G
∞
)
⊆
π
1
temp
(G)
of
π
1
temp
(G)
is
characteristic.
(ii)
(Automorphism
groups)
There
exist
natural
injective
ho-
momorphisms
Aut
|grph|
(G)
→
Aut
|grph|
(G
∞
)
,
π
1
top
(G)
→
Aut(G
∞
)
—
where
we
write
Aut
|grph|
(G
∞
)
for
the
group
of
automor-
phisms
of
G
∞
that
induce
the
identity
automorphism
of
the
underlying
semi-graph
of
G
∞
.
Moreover,
the
centralizer
of
π
1
top
(G)
in
Aut
|grph|
(G
∞
)
satisfies
the
equality
Z
Aut
|grph|
(G
∞
)
(π
1
top
(G))
=
Aut
|grph|
(G)
.
(iii)
(Action
of
the
fundamental
group
of
the
underlying
semi-graph)
Let
γ
∞
∈
π
1
top
(G)
⊆
Aut(G
∞
)
[cf.
(ii)]
be
a
generator
of
π
1
top
(G)
≃
Z.
Write
Vert(G
∞
),
Node(G
∞
),
and
Cusp(G
∞
)
for
the
sets
of
vertices,
nodes
[i.e.,
closed
edges],
and
cusps
[i.e.,
open
edges]
of
G
∞
,
respectively.
Then
there
exist
bijections
∼
∼
V
:
Z
−→
Vert(G
∞
)
,
N
:
Z
−→
Node(G
∞
)
,
∼
C
:
Z
×
Cusp(G)
−→
Cusp(G
∞
)
such
that,
for
each
a
∈
Z,
•
the
set
of
edges
that
abut
to
the
vertex
V
(a)
is
equal
to
the
disjoint
union
of
{N
(a),
N
(a
+
1)}
and
{
C(a,
z)
|
z
∈
Cusp(G)};
•
the
automorphism
of
Vert(G
∞
)
(respectively,
Node(G
∞
);
Cusp(G
∞
))
induced
by
γ
∞
∈
Aut(G
∞
)
maps
V
(a)
(respec-
tively,
N
(a);
C(a,
z))
to
V
(a
+
1)
(respectively,
N
(a
+
1);
C(a
+
1,
z)).
Combinatorial
anabelian
topics
I
77
(iv)
(Restriction
to
a
finite
sub-semi-graph)
Let
a
≤
b
∈
Z
be
integers.
Denote
by
G
[a,b]
the
[uniquely
determined]
sub-
semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
the
under-
lying
semi-graph
of
G
∞
such
that
the
set
of
vertices
of
G
[a,b]
is
equal
to
{V
(a),
V
(a
+
1),
·
·
·
,
V
(b)}
[cf.
(iii)];
denote
by
G
[a,b]
the
semi-graph
of
anabelioids
obtained
by
restricting
G
∞
to
G
[a,b]
[cf.
the
discussion
preceding
[SemiAn],
Definition
2.2].
Then
G
[a,b]
is
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-
type.
Moreover,
G
[a,a+1]
is
noncyclically
primitive.
(v)
(Restriction
to
a
sub-semi-graph
having
precisely
one
vertex)
Let
a
≤
c
≤
b
∈
Z
be
integers.
Then
the
natural
mor-
phism
of
semi-graphs
of
anabelioids
G
[c,c]
→
G
[a,b]
[cf.
(iv)]
de-
∼
termines
an
isomorphism
G
[c,c]
→
G
[a,b]
|
V
(c)
—
where
we
regard
V
(c)
∈
Vert(G
∞
)
as
a
vertex
of
G
[a,b]
.
Moreover,
if
we
write
v
∈
Vert(G)
for
the
unique
vertex
of
G
[cf.
Remark
4.1.1],
then
the
composite
of
natural
morphisms
of
semi-graphs
of
an-
abelioids
G
[c,c]
→
G
∞
→
G
determines
an
isomorphism
of
G
[c,c]
with
G|
v
.
(vi)
(Natural
isomorphisms
between
restrictions
to
finite
sub-semi-graphs)
Let
a
≤
b
∈
Z
be
integers
and
γ
∞
∈
π
1
top
(G)
⊆
Aut(G
∞
)
the
automorphism
of
G
∞
appearing
in
∼
(iii).
Then
γ
∞
determines
an
isomorphism
G
[a,b]
→
G
[a+1,b+1]
.
Proof.
First,
we
verify
assertion
(i).
To
show
that
the
natural
morphism
G
∞
→
G
induces
an
exact
sequence
1
−→
π
1
temp
(G
∞
)
−→
π
1
temp
(G)
−→
π
1
top
(G)
−→
1
,
it
suffices
to
verify
that
every
tempered
covering
of
G
∞
determines,
via
the
morphism
G
∞
→
G,
a
tempered
covering
of
G.
But
this
follows
immediately,
in
light
of
the
definition
of
a
tempered
covering,
from
the
finiteness
of
the
underlying
semi-graph
G
and
the
topologically
finitely
generated
nature
of
the
verticial
subgroups
of
the
tempered
fundamental
group
π
1
temp
(G
∞
)
of
G
∞
.
On
the
other
hand,
the
fact
that
the
subgroup
π
1
temp
(G
∞
)
⊆
π
1
temp
(G)
is
characteristic
follows
immediately
from
the
observation
that
the
quotient
π
1
temp
(G)
π
1
temp
(G)/π
1
temp
(G
∞
)
may
be
characterized
as
the
maximal
discrete
free
quotient
of
π
1
temp
(G)
[cf.
the
argument
of
[André],
Lemma
6.1.1].
This
completes
the
proof
of
asser-
tion
(i).
Next,
we
verify
assertion
(ii).
The
existence
of
a
natural
injec-
tion
π
1
top
(G)
→
Aut(G
∞
)
follows
immediately
from
the
definition
of
78
Yuichiro
Hoshi
and
Shinichi
Mochizuki
the
connected
tempered
covering
G
∞
→
G,
together
with
the
fact
that
π
1
top
(G)
is
abelian.
On
the
other
hand,
it
follows
immediately
from
assertion
(i),
together
with
the
various
definitions
involved,
that
any
element
of
Aut
|grph|
(G)
determines
—
up
to
composition
with
an
el-
ement
of
π
1
top
(G)
⊆
Aut(G
∞
)
—
an
automorphism
of
G
∞
.
There-
fore,
by
composing
with
a
suitable
element
of
π
1
top
(G)
⊆
Aut(G
∞
),
one
obtains
a
uniquely
determined
element
of
Aut
|grph|
(G
∞
),
hence
also
a
natural
injective
homomorphism
Aut
|grph|
(G)
→
Aut
|grph|
(G
∞
).
Next,
to
verify
the
equality
Z
Aut
|grph|
(G
∞
)
(π
1
top
(G))
=
Aut
|grph|
(G),
observe
that
π
1
temp
(G
∞
)
is
center-free
[cf.
[SemiAn],
Example
2.10;
[SemiAn],
Proposition
3.6,
(iv)];
this
implies
that
we
have
a
natural
isomorphism
out
π
1
temp
(G)
≃
π
1
temp
(G
∞
)
π
1
top
(G)
[cf.
the
discussion
entitled
“Topo-
logical
groups”
in
§0].
Thus,
in
light
of
the
[easily
verified]
inclusion
Aut
|grph|
(G)
⊆
Z
Aut
|grph|
(G
∞
)
(π
1
top
(G)),
the
desired
equality
follows
im-
mediately
from
[CmbGC],
Proposition
1.5,
(ii).
This
completes
the
proof
of
assertion
(ii).
Assertions
(iii),
(iv),
(v),
and
(vi)
follow
immediately
from
the
defi-
Q.E.D.
nition
of
the
connected
tempered
covering
G
∞
→
G.
Definition
4.4.
We
shall
write
Dehn(G)
=
{
α
∈
Aut
|grph|
(G)
|
α
G|
v
=
id
G|
v
for
any
v
∈
Vert(G)
}
def
—
where
we
refer
to
Definitions
2.1,
(iii);
2.14,
(ii);
Remark
2.5.1,
(ii),
concerning
“α
G|
v
”.
We
shall
refer
to
an
element
of
Dehn(G)
as
a
profinite
Dehn
multi-twist
of
G.
Proposition
4.5
(Equalities
concerning
the
group
of
profi-
nite
Dehn
multi-twists).
It
holds
that
|Π
v
|
|Π
z
|
(G)
=
(G)
Dehn(G)
=
v∈Vert(G)
Aut
z∈VCN(G)
Aut
=
z∈VCN(G)
Out
|Π
z
|
(Π
G
)
⊆
Aut
|grph|
(G)
[cf.
Definitions
2.13;
2.6,
(i);
[CmbGC],
Proposition
1.2,
(ii)]
—
where
we
use
the
notation
“Π
(−)
”
to
denote
a
VCN-subgroup
[cf.
Defini-
tion
2.1,
(i)]
of
Π
G
associated
to
“(−)”
∈
VCN(G).
Proof.
The
first
equality
follows
immediately
from
the
various
def-
initions
involved
[cf.
also
[CmbGC],
Proposition
1.2,
(i)].
The
second
Combinatorial
anabelian
topics
I
79
equality
follows
immediately
from
the
fact
that
any
edge-like
subgroup
is
contained
in
a
verticial
subgroup.
The
third
equality
follows
imme-
diately
from
Proposition
2.7,
(ii).
This
completes
the
proof
of
Proposi-
tion
4.5.
Q.E.D.
Lemma
4.6
(Construction
of
certain
homomorphisms).
Let
e
def
=
e
(G)
∈
Node(G).
Then
the
following
hold:
e
∈
Node(
G),
(i)
Let
α
∈
Dehn(G)
be
a
profinite
Dehn
multi-twist
of
G
and
Write
w
v
∈
V(
e
)
⊆
Vert(
G).
for
the
unique
element
of
the
complement
V(
e
)
\
{
v
}
[cf.
[NodNon],
Remark
1.2.1,
(iii)].
Then
there
exists
a
unique
lifting
α[
v
]
∈
Aut(Π
G
)
of
α
which
preserves
the
verticial
subgroup
Π
v
⊆
Π
G
of
Π
G
associated
to
and
induces
the
identity
automorphism
of
Π
v
.
v
∈
Vert(
G)
Moreover,
this
lifting
α[
v
]
preserves
the
verticial
subgroup
and
there
exists
∈
Vert(
G),
Π
w
⊆
Π
G
of
Π
G
associated
to
w
a
unique
element
δ
e
,
v
∈
Π
e
of
the
edge-like
subgroup
Π
e
⊆
Π
G
such
that
the
restriction
of
of
Π
G
associated
to
e
∈
Node(
G)
α[
v
]
to
Π
w
is
the
inner
automorphism
determined
by
δ
e
,
v
∈
Π
e
(⊆
Π
w
).
(ii)
For
v
∈
V(
e
),
denote
by
D
e
,
v
:
Dehn(G)
→
Λ
G
the
composite
of
the
map
Dehn(G)
−→
Π
e
given
by
assigning
α
→
δ
e
,
v
∈
Π
e
[cf.
(i)]
and
the
isomorphism
syn
b
∼
syn
v
∼
Π
e
−→
Λ
v
−→
Λ
G
def
[cf.
Corollary
3.9,
(ii),
(v)]
—
where
we
write
v
=
v
(G)
and
b
for
the
branch
of
e
determined
by
the
unique
branch
of
e
which
abuts
to
v
.
Then
the
map
D
e
,
v
:
Dehn(G)
→
Λ
G
is
a
homomorphism
of
profinite
groups
which
does
not
depend
on
the
choice
of
the
element
v
∈
V(
e
),
i.e.,
if
w
∈
V(
e
)
\
{
v
},
then
D
e
,
v
=
D
e
,
w
.
Moreover,
the
homomorphism
D
e
,
v
(=
D
e
,
w
)
depends
only
on
e
∈
Node(G),
i.e.,
it
does
not
such
that
depend
on
the
choice
of
the
element
e
∈
Node(
G)
e
(G)
=
e.
Proof.
First,
we
verify
assertion
(i).
The
fact
that
there
exists
a
unique
lifting
α[
v
]
∈
Aut(Π
G
)
of
α
which
preserves
Π
v
and
induces
the
80
Yuichiro
Hoshi
and
Shinichi
Mochizuki
identity
automorphism
of
Π
v
follows
immediately,
in
light
of
the
slimness
of
Π
v
[cf.
[CmbGC],
Remark
1.1.3]
and
the
commensurable
terminality
of
Π
v
in
Π
G
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
from
the
fact
that
α
∈
Out
|Π
v
|
(Π
G
)
[cf.
Proposition
4.5].
The
fact
that
α[
v
]
preserves
v
],
from
the
fact
Π
w
follows
immediately,
in
light
of
the
graphicity
of
α[
that
Π
w
is
the
unique
verticial
subgroup
H
of
Π
G
such
that
H
=
Π
v
and
Π
e
⊆
H
[cf.
[NodNon],
Remark
1.2.1,
(iii);
[NodNon],
Lemma
1.7],
together
with
the
fact
that
α[
v
]
preserves
Π
v
,
Π
e
⊆
Π
G
.
The
fact
that
there
exists
a
unique
element
δ
e
,
v
∈
Π
e
of
Π
e
such
that
the
restriction
of
α[
v
]
to
Π
w
is
the
inner
automorphism
determined
by
δ
e
,
v
follows
immediately,
in
light
of
the
slimness
of
Π
w
[cf.
[CmbGC],
Remark
1.1.3]
and
the
commensurable
terminality
of
Π
e
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
from
the
fact
that
α
∈
Out
|Π
w
|
(Π
G
)
[cf.
Proposition
4.5].
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
fact
that
the
map
D
e
,
v
is
a
homomorphism
follows
immediately
from
the
various
uniqueness
properties
discussed
in
assertion
(i).
The
fact
that
the
map
D
e
,
v
does
not
depend
on
the
choice
of
the
element
v
∈
V(
e
)
follows
immediately
from
Corollary
3.9,
(vi).
The
fact
that
the
homomorphism
D
e
,
v
does
not
depend
on
the
choice
of
the
element
such
that
e
(G)
=
e
follows
immediately
from
the
definition
e
∈
Node(
G)
Q.E.D.
of
the
map
D
e
,
v
.
This
completes
the
proof
of
assertion
(ii).
Definition
4.7.
For
each
node
e
∈
Node(G)
of
G,
we
shall
write
def
D
e
=
D
e
,
v
:
Dehn(G)
−→
Λ
G
for
the
homomorphism
obtained
in
Lemma
4.6,
(ii).
[Note
that
it
follows
from
Lemma
4.6,
(ii),
that
this
homomorphism
depends
only
on
e
∈
Node(G).]
We
shall
write
def
D
G
=
e∈Node(G)
D
e
:
Dehn(G)
−→
Λ
G
.
Node(G)
Theorem
4.8
(Properties
of
profinite
Dehn
multi-twists).
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
an-
abelioids
of
pro-Σ
PSC-type.
Then
the
following
hold:
(i)
(Normality)
Dehn(G)
is
normal
in
Aut(G).
Combinatorial
anabelian
topics
I
(ii)
81
(Compatibility
with
generization)
Let
S
⊆
Node(G).
Then
—
relative
to
the
inclusion
Aut
S
(G)
⊆
Aut(G
S
)
[cf.
Definition
2.8]
induced
by
the
specialization
outer
isomorphism
∼
Π
G
→
Π
G
S
with
respect
to
S
[cf.
Proposition
2.9,
(ii)]
—
we
have
a
diagram
of
inclusions
Dehn(G)
←
∩
Aut
S
(G)
Dehn(G
S
)
∩
→
Aut(G
S
)
.
Moreover,
if
we
regard
Node(G
S
)
as
a
subset
of
Node(G),
then
the
above
inclusion
Dehn(G
S
)
→
Dehn(G)
fits
into
a
commutative
diagram
of
profinite
groups
Dehn(G
S
)
−−−−→
Dehn(G)
⏐
⏐
⏐
⏐
D
D
G
S
G
−−−−→
Node(G
S
)
Λ
G
Node(G)
Λ
G
—
where
the
lower
horizontal
arrow
is
the
natural
inclusion
determined
by
the
inclusion
Node(G
S
)
→
Node(G)
and
the
∼
natural
isomorphism
Λ
G
S
→
Λ
G
[cf.
Corollary
3.9,
(i)].
(iii)
(Compatibility
with
“surgery”)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G,
S
⊆
Node(G|
H
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
sepa-
rating
type
[cf.
Definition
2.5,
(i)],
and
T
⊆
Cusp((G|
H
)
S
)
[cf.
Definition
2.5,
(ii)]
an
omittable
[cf.
Definition
2.4,
(i)]
subset
of
Cusp((G|
H
)
S
).
Then
the
natural
homomorphism
Aut
HS•T
(G)
α
−→
Aut(((G|
H
)
S
)
•T
)
→
α
((G|
H
)
S
)
•T
[cf.
Definitions
2.4,
(ii);
2.14,
(ii)]
induces
a
homomorphism
Dehn(G)
−→
Dehn(((G|
H
)
S
)
•T
)
.
Moreover,
if
we
regard
Node(((G|
H
)
S
)
•T
)
as
a
sub-
set
of
Node(G),
then
the
above
homomorphism
Dehn(G)
→
82
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Dehn(((G|
H
)
S
)
•T
)
fits
into
a
commutative
diagram
of
profi-
nite
groups
Dehn(G)
−−−−→
Dehn(((G|
H
)
S
)
•T
)
⏐
⏐
⏐
⏐
D
((G|
)
)
D
G
H
S
•T
Node(G)
Λ
G
−−−−→
Node(((G|
H
)
S
)
•T
)
Λ
G
—
where
the
lower
horizontal
arrow
is
the
natural
projection,
∼
and
we
apply
the
natural
isomorphism
Λ
G
→
Λ
((G|
H
)
S
)
•T
[cf.
Corollary
3.9,
(ii)].
(iv)
(Structure
of
the
group
of
profinite
Dehn
multi-twists)
The
homomorphism
defined
in
Definition
4.7
Λ
G
D
G
:
Dehn(G)
−→
Node(G)
is
an
isomorphism
of
profinite
groups
that
is
functorial,
in
G,
with
respect
to
isomorphisms
of
semi-graphs
of
anabe-
lioids
of
pro-Σ
PSC-type.
In
particular,
Dehn(G)
is
a
finitely
Σ
-module
of
rank
Node(G)
.
We
shall
generated
free
Z
refer
to
a
nontrivial
profinite
Dehn
multi-twist
whose
image
∈
Node(G)
Λ
G
lies
in
a
direct
summand
[i.e.,
in
a
single
“Λ
G
”]
as
a
profinite
Dehn
twist.
(v)
(Conjugation
action
on
the
group
of
profinite
Dehn
multi-twists)
The
action
of
Aut(G)
on
Node(G)
Λ
G
∼
Aut(G)
−→
Aut(Dehn(G))
−→
Aut(
Λ
G
)
Node(G)
determined
by
conjugation
by
elements
of
Aut(G)
[cf.
(i)]
and
the
isomorphism
of
(iv)
coincides
with
the
action
of
Aut(G)
on
Node(G)
Λ
G
determined
by
the
action
χ
G
of
Aut(G)
on
Λ
G
and
the
natural
action
of
Aut(G)
on
the
finite
set
Node(G).
Proof.
Assertions
(i),
(ii),
and
(iii)
follow
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(iv).
The
functo-
riality
of
the
homomorphism
D
G
follows
immediately
from
the
various
definitions
involved.
The
rest
of
the
proof
of
assertion
(iv)
is
devoted
to
verifying
that
the
homomorphism
D
G
is
an
isomorphism.
First,
we
claim
that
Combinatorial
anabelian
topics
I
83
(∗
1
):
if
G
is
noncyclically
primitive
[cf.
Definition
4.1],
then
the
homomorphism
D
G
is
injective.
Indeed,
this
follows
immediately
from
Lemma
4.2,
together
with
the
definition
of
the
homomorphism
D
G
.
This
completes
the
proof
of
the
above
claim
(∗
1
).
Next,
we
claim
that
(∗
2
):
if
G
is
cyclically
primitive
[cf.
Definition
4.1],
then
the
homomorphism
D
G
is
injective.
Indeed,
let
α
∈
Ker(D
G
)
⊆
Out(Π
G
)
be
an
element
of
Ker(D
G
).
Since
we
are
in
the
situation
of
Lemma
4.3,
we
shall
apply
the
notational
con-
ventions
established
in
Lemma
4.3.
Denote
by
α
∞
∈
Aut
|grph|
(G
∞
)
the
automorphism
of
G
∞
determined
by
α
[cf.
Lemma
4.3,
(ii)];
for
integers
a
≤
b
∈
Z,
denote
by
α
[a,b]
∈
Aut
|grph|
(G
[a,b]
)
the
automorphism
of
G
[a,b]
obtained
by
restricting
α
∞
∈
Aut
|grph|
(G
∞
).
Then
since
α
is
a
profinite
Dehn
multi-twist,
one
may
verify
easily
that
α
[a,b]
is
a
profinite
Dehn
multi-twist
of
G
[a,b]
.
Thus,
since
G
[a,a+1]
is
noncyclically
primitive
[cf.
Lemma
4.3,
(iv)],
it
follows
immediately
from
the
fact
that
α
∈
Ker(D
G
),
together
with
the
claim
(∗
1
),
that
α
[a,a+1]
is
trivial.
Moreover,
for
any
a
<
b
∈
Z,
it
follows
—
by
applying
induction
on
b
−
a
and
considering,
in
light
of
the
claim
(∗
1
),
the
various
generizations
[cf.
assertion
(ii)]
of
G
[a,b]
with
respect
to
sets
of
the
form
“Node(G
[a.b]
)
\
{e}”
—
that
the
profinite
Dehn
multi-twist
α
[a,b]
,
hence
also
the
automorphism
α
∞
,
is
trivial.
In
particular,
it
holds
that
α
is
trivial
[cf.
Lemma
4.3,
(ii)],
as
desired.
This
completes
the
proof
of
the
above
claim
(∗
2
).
Next,
we
claim
that
(∗
3
):
for
arbitrary
G,
the
homomorphism
D
G
is
injec-
tive.
We
verify
this
claim
(∗
3
)
by
induction
on
Node(G)
.
If
Node(G)
≤
1,
then
the
claim
(∗
3
)
follows
formally
from
the
claims
(∗
1
)
and
(∗
2
).
Now
suppose
that
Node(G)
>
1,
and
that
the
induction
hypothesis
is
in
force.
Let
e
∈
Node(G)
be
a
node
of
G.
Write
H
for
the
unique
sub-semi-graph
of
PSC-type
of
G
whose
set
of
vertices
is
V(e).
Then
one
may
verify
def
easily
that
S
=
Node(G|
H
)
\
{e}
is
not
of
separating
type
as
a
subset
of
Node(G|
H
).
Thus,
since
(G|
H
)
S
has
precisely
one
node,
it
follows
immediately
from
assertion
(iii),
together
with
the
claims
(∗
1
)
and
(∗
2
),
that
the
profinite
Dehn
multi-twist
α
(G|
H
)
S
of
(G|
H
)
S
determined
by
α
∈
Dehn(G)
is
trivial.
In
particular,
it
follows
immediately
from
the
def-
inition
of
a
generization
[cf.,
especially,
the
definition
of
the
anabelioids
corresponding
to
the
vertices
of
a
generization
given
in
Definition
2.8,
(vi)],
together
with
the
definition
of
a
profinite
Dehn
multi-twist,
that
84
Yuichiro
Hoshi
and
Shinichi
Mochizuki
the
automorphism
α
G
{e}
of
the
generization
G
{e}
determined
by
α
[cf.
Proposition
2.9,
(ii)]
is
a
profinite
Dehn
multi-twist.
Therefore,
since
Node(G
{e}
)
<
Node(G)
,
it
follows
immediately
from
assertion
(ii),
together
with
the
induction
hypothesis,
that
α
G
{e}
∈
Ker(D
G
{e}
),
hence
also
α
∈
Ker(D
G
),
is
trivial.
This
completes
the
proof
of
the
claim
(∗
3
).
Next,
we
claim
that
(∗
4
):
if
G
is
noncyclically
primitive
[cf.
Definition
4.1],
then
the
homomorphism
D
G
is
surjective.
Indeed,
this
follows
immediately
from
Lemma
4.2,
together
with
the
various
definitions
involved.
This
completes
the
proof
of
the
claim
(∗
4
).
Next,
we
claim
that
(∗
5
):
if
G
is
cyclically
primitive
[cf.
Definition
4.1],
then
the
homomorphism
D
G
is
surjective.
Indeed,
let
λ
∈
Λ
G
be
an
element
of
Λ
G
.
Since
we
are
in
the
situation
of
Lemma
4.3,
we
shall
apply
the
notational
conventions
established
in
Lemma
4.3.
Then
it
follows
immediately
from
Corollary
3.9,
(ii),
together
with
Lemma
4.3,
(v),
that
for
any
integers
a
≤
0
<
b
∈
Z,
the
natural
morphisms
G
[0,0]
→
G
[a,b]
and
G
[0,0]
→
G
∞
→
G
induce
∼
∼
isomorphisms
Λ
G
[a,b]
←
Λ
G
[0,0]
→
Λ
G
.
By
abuse
of
notation,
write
λ
∈
Λ
G
[a,b]
for
the
element
of
Λ
G
[a,b]
corresponding
to
λ
∈
Λ
G
.
Now
since
G
[0,1]
is
noncyclically
primitive
[cf.
Lemma
4.3,
(iv)],
it
follows
from
the
claims
(∗
1
),
(∗
4
)
that
there
exists
a
unique
profinite
Dehn
multi-twist
λ
[0,1]
∈
Dehn(G
[0,1]
)
such
that
D
G
[0,1]
(λ
[0,1]
)
=
λ.
Next,
we
claim
that
(†)
:
for
any
a
≤
0
<
b
∈
Z,
there
exists
a
[necessarily
unique
—
cf.
claim
(∗
3
)]
profinite
Dehn
multi-twist
λ
[a,b]
∈
Dehn(G
[a,b]
)
such
that
D
e
(λ
[a,b]
)
=
λ
for
every
node
e
∈
Node(G
[a,b]
).
We
verify
this
claim
(†)
by
induction
on
b−a.
If
b−a
=
1,
or
equivalently,
[a,
b]
=
[0,
1],
then
we
have
already
shown
the
existence
of
a
profinite
Dehn
multi-twist
λ
[0,1]
∈
Dehn(G
[0,1]
)
of
the
desired
type.
Now
suppose
that
1
<
b
−
a,
and
that
for
I
∈
{[a,
b
−
1],
[a
+
1,
b]},
there
exists
a
profi-
nite
Dehn
multi-twist
λ
I
∈
Dehn(G
I
)
such
that
D
e
(λ
I
)
=
λ
for
every
node
e
∈
Node(G
I
).
Then
one
may
verify
easily
that
Node(G
I
)
may
be
def
regarded
as
a
subset
of
Node(G
[a,b]
),
that
H
[a,b]
=
(G
[a,b]
)
Node(G
I
)
is
noncyclically
primitive,
and
that,
if
one
allows
v
to
range
over
the
[two]
vertices
of
H
[a,b]
,
then
the
resulting
semi-graphs
of
anabelioids
(H
[a,b]
)|
v
def
are
naturally
isomorphic
to
H
I
=
(G
I
)
Node(G
I
)
and
G
[c
I
,c
I
]
,
where
we
write
c
I
for
b
(respectively,
a)
if
I
=
[a,
b−1]
(respectively,
I
=
[a+1,
b]).
Combinatorial
anabelian
topics
I
85
Let
Π
e
I
⊆
Π
H
I
be
a
cuspidal
subgroup
of
Π
H
I
corresponding
to
the
cusp
e
I
determined
by
the
unique
node
of
H
[a,b]
;
Π
e
[cI
,cI
]
⊆
Π
G
[cI
,cI
]
a
cuspidal
subgroup
of
Π
G
[cI
,cI
]
corresponding
to
the
cusp
e
[c
I
,c
I
]
de-
I
∈
Aut(Π
H
)
a
lifting
of
the
termined
by
the
unique
node
of
H
[a,b]
;
λ
I
outomorphism
of
Π
H
I
determined
by
λ
I
∈
Dehn(G
I
)
→
Aut(H
I
)
[cf.
Proposition
2.9,
(ii)]
which
preserves
Π
e
I
and
induces
the
identity
au-
tomorphism
of
Π
e
I
.
[Note
that
since
λ
I
∈
Dehn(G
I
),
one
may
verify
I
∈
Aut(Π
H
)
exists.]
Then
for
any
element
easily
that
such
a
lifting
λ
I
δ
∈
Π
e
[cI
,cI
]
of
Π
e
[cI
,cI
]
,
it
follows
immediately
from
Lemma
4.2
that
by
∼
gluing
—
by
means
of
the
natural
isomorphism
Π
e
I
→
Π
e
[cI
,cI
]
—
the
I
∈
Aut(Π
H
)
to
the
inner
automorphism
of
Π
G
by
automorphism
λ
I
[cI
,cI
]
δ
∈
Π
e
[cI
,cI
]
,
we
obtain
an
outomorphism
λ
[a,b]
[δ]
of
Π
H
[a,b]
,
which
—
in
light
of
[CmbGC],
Proposition
1.5,
(ii),
together
with
the
fact
that
λ
I
∈
Dehn(G
I
)
—
is
contained
in
Dehn(G
[a,b]
)
⊆
Aut
|grph|
(G
[a,b]
)
→
Aut
|grph|
(H
[a,b]
)
⊆
Out(Π
H
[a,b]
)
[cf.
Proposition
2.9,
(ii)].
Now
it
follows
immediately
from
the
definition
of
the
homomorphism
“D
e
”
that
the
assignment
δ
→
D
e
G
[a,b]
(λ
a,b
[δ])
—
where
we
write
e
G
[a,b]
for
the
node
of
G
[a,b]
corresponding
to
the
unique
∼
node
of
H
[a,b]
—
determines
a
bijection
Π
e
[cI
,cI
]
→
Λ
G
.
Thus,
since
D
e
(λ
I
)
=
λ
for
every
node
e
∈
Node(G
I
),
we
conclude
that
there
exists
a
unique
element
δ
∈
Π
e
[cI
,cI
]
of
Π
e
[cI
,cI
]
such
that
D
e
(λ
[a,b]
[δ])
=
λ
for
every
node
e
∈
Node(G
[a,b]
).
This
completes
the
proof
of
the
claim
(†).
Write
λ
∞
∈
Aut
|grph|
(G
∞
)
for
the
automorphism
of
G
∞
determined
by
the
λ
[a,b]
’s
of
the
claim
(†).
Now
since
D
e
(λ
[a,b]
)
=
λ
for
arbitrary
a
<
b
∈
Z
and
e
∈
Node(G
[a,b]
),
one
may
verify
easily,
by
applying
the
claim
(∗
3
),
that
the
automorphism
λ
∞
commutes
with
the
natural
action
of
π
1
top
(G)
≃
Z
on
G
∞
.
Thus,
the
automorphism
λ
∞
determines
an
automorphism
λ
G
∈
Aut
|grph|
(G)
of
G
[cf.
Lemma
4.3,
(ii)].
Moreover,
it
follows
immediately
from
the
definition
of
λ
G
,
together
with
the
fact
that
D
e
(λ
[a,b]
)
=
λ
for
arbitrary
a
<
b
∈
Z
and
e
∈
Node(G
[a,b]
),
that
λ
G
is
a
profinite
Dehn
multi-twist
such
that
D
G
(λ
G
)
=
λ
∈
Λ
G
.
This
completes
the
proof
of
the
claim
(∗
5
).
Finally,
we
claim
that
(∗
6
):
for
arbitrary
G,
the
homomorphism
D
G
is
sur-
jective.
86
Yuichiro
Hoshi
and
Shinichi
Mochizuki
For
each
node
e
∈
Node(G)
of
G,
it
follows
from
assertion
(ii)
that
we
have
a
commutative
diagram
of
profinite
groups
Dehn(G
Node(G)\{e}
)
−−−−→
⏐
D
G
Node(G)\{e}
⏐
Λ
G
−−−−→
Dehn(G)
⏐
⏐
D
G
e
∈Node(G)
Λ
G
—
where
the
lower
horizontal
arrow
is
the
natural
inclusion
into
the
component
indexed
by
e.
Now
since
Node(G
Node(G)\{e}
)
=
1,
it
follows
from
the
claims
(∗
4
),
(∗
5
)
that
the
left-hand
vertical
arrow
D
G
Node(G)\{e}
in
the
above
commutative
diagram
is
surjective.
Therefore,
by
allowing
“e”
to
vary
among
the
elements
of
Node(G),
we
conclude
that
D
G
is
surjective.
This
completes
the
proof
of
the
claim
(∗
6
)
—
hence
also,
in
light
of
the
claim
(∗
3
)
—
of
assertion
(iv).
Finally,
assertion
(v)
follows
immediately
from
the
various
defini-
tions
involved,
together
with
assertion
(iv).
This
completes
the
proof
of
Theorem
4.8.
Q.E.D.
Remark
4.8.1.
In
the
notation
of
Theorem
4.8,
denote
by
π
1
temp
(G)
the
tempered
fundamental
group
of
G
[cf.
the
discussion
pre-
ceding
[SemiAn],
Proposition
3.6],
by
π
1
top
(G)
the
[discrete]
topological
fundamental
group
of
the
underlying
semi-graph
G
of
G,
by
G
∞
→
G
the
connected
tempered
covering
of
G
corresponding
to
the
natural
sur-
jection
π
1
temp
(G)
π
1
top
(G)
[where
we
refer
to
[SemiAn],
§3,
concerning
tempered
coverings
of
a
semi-graph
of
anabelioids],
by
Aut
|grph|
(G
∞
)
the
group
of
automorphisms
of
G
∞
that
induce
the
identity
automorphism
of
the
underlying
semi-graph
of
G
∞
,
and
by
Dehn(G
∞
)
⊆
Aut
|grph|
(G
∞
)
the
group
of
“profinite
Dehn
multi-twists”
of
G
∞
—
i.e.,
automorphisms
of
G
∞
which
induce
the
identity
automorphism
on
the
underlying
semi-
graph
of
G
∞
,
as
well
as
on
the
anabelioids
of
G
∞
corresponding
to
the
vertices
of
G
∞
.
Then
the
following
hold:
(i)
The
natural
morphism
G
∞
→
G
induces
an
exact
sequence
1
−→
π
1
temp
(G
∞
)
−→
π
1
temp
(G)
−→
π
1
top
(G)
−→
1
.
Moreover,
the
subgroup
π
1
temp
(G
∞
)
⊆
π
1
temp
(G)
of
π
1
temp
(G)
is
characteristic.
(ii)
There
exist
natural
injections
Aut
|grph|
(G)
→
Aut
|grph|
(G
∞
)
,
Dehn(G)
→
Dehn(G
∞
)
,
Combinatorial
anabelian
topics
I
87
π
1
top
(G)
→
Aut(G
∞
)
—
where
the
third
injection
is
determined
up
to
composition
with
a
π
1
top
(G)-inner
automorphism
—
which
satisfy
the
equal-
ities
Z
Aut
|grph|
(G
∞
)
(π
1
top
(G))
=
Aut
|grph|
(G)
;
Dehn(G)
=
Aut
|grph|
(G)
∩
Dehn(G
∞
)
.
(iii)
There
exists
a
natural
isomorphism
∼
Dehn(G
∞
)
→
Λ
G
.
Node(G
∞
)
Indeed,
assertion
(i)
(respectively,
(ii))
follows
immediately
from
a
simi-
lar
argument
to
the
argument
used
in
the
proof
of
Lemma
4.3,
(i)
(respec-
tively,
Lemma
4.3,
(ii)),
together
with
the
various
definitions
involved.
On
the
other
hand,
the
existence
of
the
natural
isomorphism
asserted
in
assertion
(iii)
follows
immediately
from
the
fact
that
the
various
ho-
momorphisms
D
(G
∞
)|
H
—
where
H
ranges
over
the
sub-semi-graphs
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
the
underlying
semi-graph
of
G
∞
,
and
we
write
(G
∞
)|
H
for
the
semi-graph
of
anabelioids
obtained
by
re-
stricting
G
∞
to
H
[cf.
the
discussion
preceding
[SemiAn],
Definition
2.2],
which
[as
is
easily
verified]
is
of
pro-Σ
PSC-type
—
are
isomorphisms.
[Note
that
since
(G
∞
)|
H
is
of
pro-Σ
PSC-type,
the
fact
that
D
(G
∞
)|
H
is
an
isomorphism
is
a
consequence
of
Theorem
4.8,
(iv).
However,
since
H
is
a
tree,
it
follows
from
the
simple
structure
of
H
that
one
may
verify
that
D
(G
∞
)|
H
is
an
isomorphism
in
a
fairly
direct
fashion,
by
arguing
as
in
the
proofs
of
the
claims
(∗
1
),
(∗
4
)
that
appear
in
the
proof
of
Theorem
4.8,
(iv).]
In
particular,
it
follows
immediately
from
assertions
(ii),
(iii)
that
one
may
recover
the
natural
isomorphism
∼
∼
Λ
G
Dehn(G)
→
Z
Node(G
)
Λ
G
(π
1
top
(G))
→
∞
Node(G)
of
Theorem
4.8,
(iv).
Definition
4.9.
We
shall
write
Aut
|grph|
(G|
v
)
Glu(G)
⊆
v∈Vert(G)
88
Yuichiro
Hoshi
and
Shinichi
Mochizuki
for
the
[closed]
subgroup
of
“glueable”
collections
of
outomorphisms
of
the
direct
product
v∈Vert(G)
Aut
|grph|
(G|
v
)
consisting
of
elements
(α
v
)
v∈Vert(G)
such
that
χ
v
(α
v
)
=
χ
w
(α
w
)
[cf.
Definition
3.8,
(ii)]
for
any
v,
w
∈
Vert(G).
Proposition
4.10
(Properties
of
automorphisms
that
fix
the
underlying
semi-graph).
(i)
(Factorization)
The
natural
homomorphism
Aut
|grph|
(G)
α
−→
→
v∈Vert(G)
Aut
|grph|
(G|
v
)
(α
G|
v
)
v∈Vert(G)
[cf.
Definition
2.14,
(ii);
Remark
2.5.1,
(ii)]
factors
through
the
closed
subgroup
Glu(G)
⊆
v∈Vert(G)
Aut
|grph|
(G|
v
).
(ii)
(Exact
sequence
relating
profinite
Dehn
multi-twists
and
glueable
outomorphisms)
The
resulting
homomor-
:
Aut
|grph|
(G)
→
Glu(G)
[cf.
(i)]
fits
into
an
exact
phism
ρ
Vert
G
sequence
of
profinite
groups
ρ
Vert
G
1
−→
Dehn(G)
−→
Aut
|grph|
(G)
−→
Glu(G)
−→
1
.
(iii)
(Surjectivity
of
cyclotomic
characters)
The
restriction
of
the
pro-Σ
cyclotomic
character
χ
G
of
G
[cf.
Definition
3.8,
(ii)]
to
Aut
|grph|
(G)
⊆
Aut(G)
Σ
)
∗
χ
G
|
Aut
|grph|
(G)
:
Aut
|grph|
(G)
−→
(
Z
—
hence
also
χ
G
—
is
surjective.
(iv)
(Liftability
of
automorphisms)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
G
and
S
⊆
Node(G|
H
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
separating
type
[cf.
Definition
2.5,
(i)].
Then
the
homo-
morphism
Aut
|grph|
(G)
α
−→
Aut
|grph|
((G|
H
)
S
)
→
α
(G|
H
)
S
[cf.
Definitions
2.5,
(ii);
2.14,
(ii)]
is
surjective.
Combinatorial
anabelian
topics
I
89
Proof.
Assertion
(i)
follows
immediately
from
Corollary
3.9,
(iv).
Next,
we
verify
assertion
(ii).
It
follows
immediately
from
the
various
)
=
Dehn(G)
⊆
Aut
|grph|
(G).
Thus,
definitions
involved
that
Ker(ρ
Vert
G
to
complete
the
proof
of
assertion
(ii),
it
suffices
to
verify
that
the
ho-
is
surjective.
momorphism
ρ
Vert
G
Now
we
claim
that
(∗
1
):
if
G
is
noncyclically
primitive
[cf.
Definition
4.1],
is
surjective.
then
the
homomorphism
ρ
Vert
G
Indeed,
this
follows
immediately
from
Corollary
3.9,
(v);
Lemma
4.2,
together
with
the
various
definitions
involved.
This
completes
the
proof
of
the
claim
(∗
1
).
Next,
we
claim
that
(∗
2
):
if
G
is
cyclically
primitive
[cf.
Definition
4.1],
is
surjective.
then
the
homomorphism
ρ
Vert
G
Indeed,
since
we
are
in
the
situation
of
Lemma
4.3,
we
shall
apply
the
notational
conventions
established
in
Lemma
4.3.
Then
it
follows
im-
mediately
from
the
fact
that
Vert(G)
=
1
[cf.
Remark
4.1.1],
together
with
Lemma
4.3,
(v),
that
the
composite
of
natural
morphisms
G
[0,0]
→
∼
G
∞
→
G
determines
a
natural
identification
Glu(G)
→
Aut
|grph|
(G
[0,0]
).
∼
∼
Let
α
=
α
[0,0]
∈
Glu(G)
→
Aut
|grph|
(G
[0,0]
)
be
an
element
of
Glu(G)
→
Aut
|grph|
(G
[0,0]
).
For
each
a
∈
Z,
denote
by
α
[a,a]
∈
Aut(G
[a,a]
)
the
au-
tomorphism
of
G
[a,a]
determined
by
conjugating
the
automorphism
α
of
∼
a
:
G
[0,0]
→
G
[a,a]
[cf.
Lemma
4.3,
(iii),
(vi)].
G
[0,0]
by
the
isomorphism
γ
∞
Then
for
any
c
<
b
∈
Z,
it
follows
from
the
various
definitions
involved
that
the
various
α
[a,a]
’s
satisfy
the
gluing
condition
necessary
to
apply
the
claim
(∗
1
),
hence
that
we
may
glue
them
together
[cf.
the
proof
of
the
claim
(∗
3
)
below
for
more
details
concerning
this
sort
of
gluing
argu-
ment]
to
obtain
a(n)
[not
necessarily
unique]
element
of
Aut
|grph|
(G
[c,b]
).
Thus,
by
allowing
c
<
b
∈
Z
to
vary,
we
obtain
a(n)
[not
necessarily
unique]
element
α
∞
∈
Aut
|grph|
(G
∞
)
of
Aut
|grph|
(G
∞
).
Now
it
follows
immediately
from
the
definition
of
α
∞
that
for
any
γ
∈
π
1
top
(G),
the
def
−1
·
γ
−1
of
G
∞
is
a
“profinite
Dehn
automorphism
[α
∞
,
γ]
=
α
∞
·
γ
·
α
∞
multi-twist”
of
G
∞
,
i.e.,
[α
∞
,
γ]
∈
Dehn(G
∞
)
[cf.
Remark
4.8.1].
More-
over,
one
may
verify
easily
that
the
assignment
γ
→
[α
∞
,
γ]
determines
a
1-cocycle
π
1
top
(G)
→
Dehn(G
∞
).
Thus,
by
Remark
4.8.1,
(iii),
together
with
the
[easily
verified]
fact
that
H
1
(Z,
Z
Σ
)
=
{0}
Z
90
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Σ
—
where
we
take
the
action
of
Z
on
Z
Z
to
be
the
action
determined
Σ
and
the
action
of
Z
on
the
index
set
by
the
trivial
action
of
Z
on
Z
Z
given
by
addition
—
we
conclude
that
there
exists
an
element
β
∈
Dehn(G
∞
)
such
that
the
automorphism
β
◦
α
∞
commutes
with
the
nat-
ural
action
of
π
1
top
(G)
on
G
∞
.
In
particular,
it
follows
from
Lemma
4.3,
(ii),
that
β◦α
∞
determines
an
element
α
G
∈
Aut
|grph|
(G)
of
Aut
|grph|
(G).
Now
since
β
∈
Dehn(G
∞
),
it
follows
immediately
from
the
various
def-
∼
initions
involved
that
ρ
Vert
(α
G
)
=
α
∈
Glu(G)
→
Aut
|grph|
(G
[0,0]
).
This
G
completes
the
proof
of
the
claim
(∗
2
).
Finally,
we
claim
that
is
sur-
(∗
3
):
for
arbitrary
G,
the
homomorphism
ρ
Vert
G
jective.
We
verify
this
claim
(∗
3
)
by
induction
on
Node(G)
.
If
Node(G)
≤
1,
then
this
follows
immediately
from
the
claims
(∗
1
),
(∗
2
).
Now
sup-
pose
that
Node(G)
>
1,
and
that
the
induction
hypothesis
is
in
force.
Let
e
∈
Node(G)
be
a
node
of
G.
Write
H
for
the
unique
sub-semi-
graph
of
PSC-type
of
G
whose
set
of
vertices
is
V(e).
Then
one
may
def
verify
easily
that
S
=
Node(G|
H
)
\
{e}
is
not
of
separating
type
as
a
subset
of
Node(G|
H
).
Thus,
since
(G|
H
)
S
has
precisely
one
node,
and
(α
v
)
v∈V(e)
may
be
regarded
as
an
element
of
Glu((G|
H
)
S
),
it
fol-
lows
from
the
claims
(∗
1
),
(∗
2
)
that
there
exists
an
automorphism
β
∈
Aut
|grph|
((G|
H
)
S
)
of
(G|
H
)
S
such
that
ρ
Vert
(G|
H
)
S
(β)
=
(α
v
)
v∈V(e)
∈
Glu((G|
H
)
S
).
Write
β
{e}
∈
Aut
|grph|
(((G|
H
)
S
)
{e}
)
for
the
auto-
morphism
of
((G|
H
)
S
)
{e}
determined
by
β
∈
Aut
|grph|
((G|
H
)
S
)
[cf.
Proposition
2.9,
(ii)].
Then
it
follows
immediately
from
Corollary
3.9,
(i),
together
with
the
definition
of
a
generization
[cf.,
especially,
the
def-
inition
of
the
anabelioids
corresponding
to
the
vertices
of
a
generization
given
in
Definition
2.8,
(vi)],
that
the
element
γ
def
=
∈
(β
{e}
,
(α
v
)
v
∈V(e)
)
Aut
|grph|
(((G|
H
)
S
)
{e}
)
×
v
∈V(e)
Aut
|grph|
(G|
v
)
may
be
regarded
as
an
element
of
Glu(G
{e}
).
Now
since
Node(G
{e}
)
<
Node(G)
,
it
follows
from
the
induction
hypothesis
that
there
ex-
ists
an
automorphism
α
{e}
∈
Aut
|grph|
(G
{e}
)
of
G
{e}
such
that
ρ
Vert
G
{e}
(α
{e}
)
=
γ
∈
Glu(G
{e}
).
On
the
other
hand,
since
β
{e}
arises
from
an
element
β
of
Aut
|grph|
((G|
H
)
S
),
it
follows
immediately
from
[CmbGC],
Proposition
1.5,
(ii),
that
α
{e}
∈
Aut
|grph|
(G
{e}
)
is
contained
in
the
image
of
Aut
|grph|
(G)
→
Aut
|grph|
(G
{e}
)
[cf.
Proposi-
tion
2.9,
(ii)].
Moreover,
since
ρ
Vert
(G|
H
)
S
(β)
=
(α
v
)
v∈V(e)
∈
Glu((G|
H
)
S
),
Combinatorial
anabelian
topics
I
91
it
follows
immediately
from
our
original
characterization
of
α
{e}
that
(α
{e}
)
=
(α
v
)
v∈Vert(G)
∈
Glu(G).
Thus,
we
conclude
that
ρ
Vert
is
ρ
Vert
G
G
surjective,
as
desired.
This
completes
the
proof
of
the
claim
(∗
3
),
hence
also
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
First,
let
us
observe
that
one
may
verify
easily
that
there
exist
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-
type
H
that
is
totally
degenerate
[cf.
Definition
2.3,
(iv)],
a
subset
S
⊆
∼
Node(H),
and
an
isomorphism
of
semi-graphs
of
anabelioids
H
S
→
|grph|
G.
Now
since
we
have
a
natural
injection
Aut
(H)
→
∼
|grph|
|grph|
Aut
(H
S
)
→
Aut
(G)
[cf.
Proposition
2.9,
(ii)],
it
follows
immediately
from
Corollary
3.9,
(i),
that
to
verify
assertion
(iii),
by
replacing
G
by
H,
we
may
assume
without
loss
of
generality
that
G
is
totally
degenerate.
On
the
other
hand,
it
follows
immediately
from
as-
sertion
(ii),
together
with
Corollary
3.9,
(ii),
that
to
verify
assertion
(iii),
it
suffices
to
verify
the
surjectivity
of
χ
G|
v
for
each
v
∈
Vert(G).
Thus,
to
verify
assertion
(iii),
by
replacing
G
by
G|
v
,
we
may
assume
without
loss
of
generality
that
G
is
of
type
(0,
3)
[cf.
Definition
2.3,
(i)].
But
assertion
(iii)
in
the
case
where
G
is
of
type
(0,
3)
follows
immediately
by
consid-
ering
the
natural
outer
action
of
the
absolute
Galois
group
Gal(Q/Q)
of
the
field
of
rational
numbers
Q
—
where
we
use
the
notation
Q
to
denote
an
algebraic
closure
of
Q
—
on
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
associated
to
the
tripod
P
1
Q
\
{0,
1,
∞}
over
Q.
This
completes
the
proof
of
assertion
(iii).
def
Finally,
we
verify
assertion
(iv).
Write
H
=
(G|
H
)
S
.
Then
it
follows
immediately
from
assertion
(ii),
together
with
Theorem
4.8,
(iii),
that
the
homomorphism
Aut
|grph|
(G)
→
Aut
|grph|
(H)
in
question
fits
into
a
commutative
diagram
of
profinite
groups
ρ
Vert
G
1
−−−−→
Dehn(G)
−−−−→
Aut
|grph|
(G)
−−−−→
Glu(G)
−−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
ρ
Vert
H
−→
Glu(H)
−−−−→
1
1
−−−−→
Dehn(H)
−−−−→
Aut
|grph|
(H)
−−−
—
where
the
horizontal
sequences
are
exact.
Now
since
the
left-hand
ver-
tical
arrow
is
surjective
[cf.
Theorem
4.8,
(iii),
(iv)],
to
verify
assertion
(iv),
it
suffices
to
verify
the
surjectivity
of
the
right-hand
vertical
arrow.
But
this
follows
immediately
from
assertion
(iii),
together
with
the
defi-
nition
of
“Glu(−)”.
This
completes
the
proof
of
assertion
(iv).
Q.E.D.
92
§5.
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Comparison
with
scheme
theory
In
the
present
§,
we
discuss
[cf.
Proposition
5.6;
Theorem
5.7;
Corol-
laries
5.9,
5.10
below]
the
relationship
between
intrinsic,
group-theoretic
properties
of
profinite
Dehn
multi-twists
[such
as
length,
nondegener-
acy,
and
positive
definiteness
—
cf.
Definitions
5.1;
5.8,
(ii),
(iii)
below]
and
scheme-theoretic
characterizations
of
properties
of
outer
representa-
tions
of
pro-Σ
PSC-type
[such
as
length,
strict
nodal
nondegeneracy,
and
IPSC-ness
—
cf.
Definition
5.3,
(ii)
below;
[NodNon],
Definition
2.4,
(i),
(iii)].
The
resulting
theory
leads
naturally
to
a
proof
of
the
graphicity
of
C-admissible
outomorphisms
contained
in
the
commensurator
of
the
group
of
profinite
Dehn
multi-twists
[cf.
Theorem
5.14
below].
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Write
G
for
the
underlying
semi-graph
of
G,
Π
G
for
the
[pro-Σ]
fundamental
group
of
G,
and
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Definition
5.1.
Let
ρ
:
I
→
Aut(G)
(⊆
Out(Π
G
))
be
an
outer
rep-
resentation
of
pro-Σ
PSC-type
[cf.
[NodNon],
Definition
2.1,
(i)]
which
an
is
of
NN-type
[cf.
[NodNon],
Definition
2.4,
(iii)]
and
e
∈
Node(
G)
out
Write
Π
I
def
=
Π
G
I
[cf.
the
discussion
entitled
element
of
Node(
G).
“Topological
groups”
in
§0];
v
,
w
∈
Vert(G)
for
the
two
distinct
elements
such
that
V(
of
Vert(
G)
e
)
=
{
v
,
w}
[cf.
[NodNon],
Remark
1.2.1,
(iii)];
I
e
,
I
v
,
I
w
⊆
Π
I
for
the
inertia
subgroups
of
Π
I
associated
to
e
,
v
,
w,
respectively,
i.e.,
the
centralizers
of
Π
e
,
Π
v
,
Π
w
⊆
Π
I
in
Π
I
,
respectively
[cf.
[NodNon],
Definition
2.2].
Then
it
follows
from
condition
(3)
of
[NodNon],
Definition
2.4,
that
the
natural
homomorphism
I
v
×
I
w
→
I
e
is
an
open
injection.
Write
def
e,
ρ)
=
[I
e
:
I
v
×
I
w
]
lng
Σ
G
(
for
the
index
of
I
v
×
I
w
in
I
e
;
we
shall
refer
to
lng
Σ
e,
ρ)
as
the
Σ-length
G
(
of
e
with
respect
to
ρ.
Note
that
it
follows
immediately
from
the
various
definitions
involved
that
the
Σ-length
of
e
with
respect
to
ρ
depends
only
def
on
e
=
e
(G)
∈
Node(G)
and
ρ.
Write
def
Σ
e,
ρ)
;
lng
Σ
G
(e,
ρ)
=
lng
G
(
we
shall
refer
to
lng
Σ
G
(e,
ρ)
as
the
Σ-length
of
e
∈
Node(G)
with
respect
to
ρ.
Combinatorial
anabelian
topics
I
93
Lemma
5.2
(Outer
representations
of
SVA-type
and
profi-
nite
Dehn
multi-twists).
Let
ρ
:
I
→
Aut(G)
(⊆
Out(Π
G
))
be
an
outer
representation
of
pro-Σ
PSC-type
which
is
of
SVA-type
[cf.
[NodNon],
an
element
of
Node(
G).
Write
Definition
2.4,
(ii)]
and
e
∈
Node(
G)
def
out
Π
I
=
Π
G
I
[cf.
the
discussion
entitled
“Topological
groups”
in
for
the
two
distinct
elements
of
Vert(
G)
such
that
§0];
v
,
w
∈
Vert(
G)
V(
e
)
=
{
v
,
w}
[cf.
[NodNon],
Remark
1.2.1,
(iii)];
I
e
,
I
v
,
I
w
⊆
Π
I
for
def
the
inertia
subgroups
of
Π
I
associated
to
e
,
v
,
w,
respectively;
e
=
e
(G);
def
v
=
v
(G).
Then
the
following
hold:
(i)
(Outer
representations
of
SVA-type
and
profinite
Dehn
multi-twists)
The
outer
representation
ρ
factors
through
the
closed
subgroup
Dehn(G)
⊆
Aut(G).
By
abuse
of
notation,
write
ρ
for
the
resulting
homomorphism
I
→
Dehn(G).
(ii)
(Outer
representations
of
SVA-type
and
homomor-
phisms
of
Dehn
coordinates)
The
natural
inclusions
I
v
,
I
w
→
I
e
and
the
composite
I
e
→
Π
I
I
determine
a
diagram
of
profinite
groups
I
v
×
I
w
⏐
⏐
1
−−−−→
Π
e
−−−−→
I
e
−−−−→
I
−−−−→
1
—
where
the
lower
horizontal
sequence
is
exact,
and
the
closed
subgroups
I
v
,
I
w
⊆
I
e
determine
sections
of
the
surjection
I
e
I,
respectively
—
hence
also
homomorphisms
syn
b
v
∼
∼
∼
syn
v
∼
I
←
I
v
→
I
e
/I
w
←
Π
e
=
Π
e
→
Λ
v
→
Λ
G
∼
—
where
the
first
“
←”
denotes
the
isomorphism
given
by
the
composite
I
v
→
Π
I
I,
and
b
v
denotes
the
branch
of
e
deter-
mined
by
the
[unique]
branch
of
e
that
abuts
to
v
.
Moreover,
the
composite
of
these
homomorphisms
I
→
Λ
G
coincides
with
the
composite
ρ
D
e
I
−→
Dehn(G)
−→
Λ
G
[cf.
(i);
Definition
4.7].
In
particular,
if
ρ
is
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)],
then
the
image
of
the
94
Yuichiro
Hoshi
and
Shinichi
Mochizuki
ρ
D
composite
I
→
Dehn(G)
→
e
Λ
G
coincides
with
lng
Σ
G
(e,
ρ)
·
Λ
G
⊆
Λ
G
.
(iii)
(Centralizers
and
cyclotomic
characters)
Suppose
that
ρ
is
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)].
Let
e
∈
Node(G)
be
a
node
of
G.
Then
χ
G
(α)
=
1
[cf.
Defini-
tion
3.8,
(ii)]
for
any
α
∈
Z
Aut
{e}
(G)
(Im(ρ))
⊆
Aut
{e}
(G)
[cf.
Definition
2.6,
(i)].
Proof.
Assertion
(i)
follows
immediately
from
condition
(2
)
of
[NodNon],
Definition
2.4.
Next,
we
verify
assertion
(ii).
The
fact
that
the
natural
inclusions
I
v
,
I
w
→
I
e
and
the
composite
I
e
→
Π
I
I
give
rise
to
the
diagram
and
homomorphisms
of
the
first
and
second
displays
in
the
statement
of
assertion
(ii)
follows
immediately
from
[NodNon],
Lemma
2.5,
(iv);
condition
(2
)
of
[NodNon],
Definition
2.4.
On
the
other
hand,
it
follows
immediately
from
the
various
definitions
involved
∼
that
the
image
of
each
β
∈
I
via
the
composite
of
I
←
I
v
with
the
v
]”
of
action
I
v
→
Aut(Π
G
)
given
by
conjugation
coincides
with
the
“α[
Lemma
4.6,
(i),
in
the
case
where
one
takes
“α”
to
be
ρ(β).
Thus,
it
follows
immediately
from
the
definition
of
I
w
that
the
image
of
β
∈
I
∼
∼
via
the
composite
I
←
I
v
→
I
e
/I
w
→
Π
e
coincides
with
the
“δ
e
,
v
”
of
Lemma
4.6,
(i),
in
the
case
where
one
takes
“α”
to
be
ρ(β).
Therefore,
it
follows
immediately
from
the
definition
of
D
e
that
the
homomorphisms
of
the
final
two
displays
of
assertion
(ii)
coincide.
Thus,
the
final
portion
of
assertion
(ii)
concerning
ρ
of
SNN-type
follows
immediately
from
the
definition
of
Σ-length.
This
completes
the
proof
of
assertion
(ii).
To
ver-
ify
assertion
(iii),
let
us
first
observe
that,
by
Theorem
4.8,
(v),
the
con-
∼
jugation
action
of
α
∈
Aut
{e}
(G)
on
the
Λ
G
⊆
Node(G)
Λ
G
←
Dehn(G)
indexed
by
e
∈
Node(G)
is
given
by
multiplication
by
χ
G
(α).
On
the
other
hand,
since
N
lng
Σ
G
(e,
ρ)
=
0,
it
follows
from
the
final
portion
of
assertion
(ii)
that
the
projection
of
Im(ρ)
to
the
coordinate
indexed
by
e
is
open.
Thus,
the
fact
that
α
lies
in
the
centralizer
Z
Aut
{e}
(G)
(Im(ρ))
implies
that
χ
G
(α)
=
1,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Q.E.D.
Definition
5.3.
Let
R
be
a
complete
discrete
valuation
ring
whose
residue
field
is
separably
closed
of
characteristic
∈
Σ;
π
∈
R
a
prime
element
of
R;
v
R
the
discrete
valuation
of
R
such
that
v
R
(π)
=
1;
def
S
log
the
log
scheme
obtained
by
equipping
S
=
Spec
R
with
the
log
structure
defined
by
the
maximal
ideal
(π)
⊆
R
of
R;
s
log
the
log
scheme
Combinatorial
anabelian
topics
I
95
obtained
by
equipping
the
spectrum
s
of
the
residue
field
of
R
with
the
log
structure
induced
by
the
log
structure
of
S
log
via
the
natural
closed
immersion
s
→
S;
X
log
a
stable
log
curve
over
S
log
;
G
X
log
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
determined
by
the
special
def
fiber
X
s
log
=
X
log
×
S
log
s
log
of
the
stable
log
curve
X
log
[cf.
[CmbGC],
Σ
)
the
maximal
pro-Σ
completion
of
the
log
Example
2.5];
I
S
log
(≃
Z
fundamental
group
π
1
(S
log
)
of
S
log
.
(i)
One
may
verify
easily
that
the
natural
outer
representation
I
S
log
→
Aut(G
X
log
)
associated
to
the
stable
log
curve
X
log
over
S
log
factors
through
Dehn(G
X
log
)
⊆
Aut(G
X
log
).
We
shall
write
ρ
X
s
log
:
I
S
log
−→
Dehn(G
X
log
)
for
the
resulting
homomorphism.
(ii)
It
follows
from
the
well-known
local
structure
of
a
stable
log
curve
in
a
neighborhood
of
a
node
that
for
each
node
e
of
the
special
fiber
of
X
log
,
there
exists
a
nonzero
element
a
e
=
0
of
X,e
of
the
maximal
ideal
(π)
⊆
R
such
that
the
completion
O
the
local
ring
O
X,e
at
e
is
isomorphic
to
R[[s
1
,
s
2
]]/(s
1
s
2
−
a
e
)
—
where
s
1
,
s
2
denote
indeterminates.
Write
Σ
Σ
lng
X
log
(e)
=
v
R
(a
e
);
lng
Σ
X
log
(e)
=
[
Z
:
lng
X
log
(e)
·
Z
].
def
def
We
shall
refer
to
lng
X
log
(e)
as
the
length
of
e
and
to
lng
Σ
X
log
(e)
as
the
Σ-length
of
e.
One
verifies
easily
that
lng
X
log
(e),
hence
also
lng
Σ
X
log
(e),
depends
only
on
e,
i.e.,
is
independent
of
the
X,e
≃
R[[s
1
,
s
2
]]/(s
1
s
2
−
a
e
).
choice
of
the
isomorphism
O
Lemma
5.4
(Local
geometric
universal
outer
representa-
tions).
In
the
notation
of
Definition
5.3,
suppose
that
G
X
log
is
of
type
def
(g,
r)
[cf.
Definition
2.3,
(i);
Remark
2.3.1].
Write
N
=
Node(G
X
log
)
log
and
σ
log
:
S
log
→
(M
g,r
)
S
[cf.
the
discussion
entitled
“Curves”
in
§0]
for
the
classifying
morphism
of
the
stable
log
curve
X
log
over
S
log
.
Then
the
following
hold:
(i)
(Local
structure
of
the
moduli
stack
of
pointed
sta-
for
the
completion
of
the
local
ring
of
ble
curves)
Write
O
96
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(M
g,r
)
S
at
the
image
of
the
closed
point
of
S
via
the
underly-
ing
(1-)morphism
of
stacks
σ
of
σ
log
and
T
log
for
the
def
with
the
log
[fs]
log
scheme
obtained
by
equipping
T
=
Spec
O
log
structure
induced
by
the
log
structure
of
(M
g,r
)
S
.
[Thus,
we
have
a
tautological
strict
[cf.
[Illu],
1.2]
(1-)morphism
T
log
→
log
(M
g,r
)
S
.]
Then
there
exists
an
isomorphism
of
R-algebras
∼
R[[t
1
,
·
·
·
,
t
3g−3+r
]]
→
O
such
that
the
following
hold:
•
The
log
structure
of
the
log
scheme
T
log
is
given
by
the
following
chart:
∼
e∈Node(G
)
N
e
−→
R[[t
1
,
·
·
·
,
t
3g−3+r
]]
→
O
X
log
(n
e
1
,
·
·
·
,
n
e
N
)
→
n
n
e
t
1
e
1
·
·
·
t
N
N
—
where
we
write
N
e
for
the
copy
of
N
indexed
by
e
∈
Node(G
X
log
).
•
(ii)
→
R
For
1
≤
i
≤
N
,
the
homomorphism
of
R-algebras
O
induced
by
the
morphism
σ
maps
t
i
to
a
e
i
[cf.
Defini-
tion
5.3,
(ii)].
(Log-scheme-theoretic
description
of
log
fundamental
groups)
Write
I
T
log
for
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
π
1
(T
log
)
of
T
log
.
Then
we
have
natural
isomorphisms
∼
Σ
(1)
;
I
S
log
→
Hom
N
gp
,
Z
I
T
log
gp
Σ
Hom
N
,
Z
(1)
e∈Node(G
log
)
e
X
∼
gp
Σ
→
e∈Node(G
)
Hom
N
e
,
Z
(1)
,
∼
→
X
log
and
the
homomorphism
I
S
log
→
I
T
log
induced
by
the
classify-
ing
morphism
σ
log
is
the
homomorphism
obtained
by
applying
Σ
(1))”
to
the
homomorphism
of
the
functor
“Hom
Z
Σ
((−)
gp
,
Z
monoids
N
e∈Node(G
log
)
N
e
−→
X
.
N
(n
e
1
,
·
·
·
,
n
e
N
)
→
i=1
n
e
i
lng
X
log
(e
i
)
(iii)
(Local
geometric
universal
outer
representations)
The
natural
outer
representation
I
T
log
→
Aut(G
X
log
)
associ-
ated
to
the
stable
log
curve
over
T
log
determined
Combinatorial
anabelian
topics
I
97
log
by
the
tautological
strict
morphism
T
log
→
(M
g,r
)
S
factors
through
Dehn(G
X
log
)
⊆
Aut(G
X
log
);
thus,
we
have
a
homomor-
phism
I
T
log
→
Dehn(G
X
log
).
Moreover,
the
homomorphism
ρ
X
s
log
:
I
S
log
→
Dehn(G
X
log
)
factors
as
the
composite
of
the
ho-
momorphism
I
S
log
→
I
T
log
induced
by
σ
log
and
this
homomor-
phism
I
T
log
→
Dehn(G
X
log
).
Proof.
Assertion
(i)
follows
immediately
from
the
well-known
local
log
structure
of
the
log
stack
(M
g,r
)
S
[cf.
[Knud],
Theorem
2.7].
Assertion
(ii)
follows
immediately
from
assertion
(i),
together
with
the
well-known
structure
of
the
log
fundamental
groups
of
S
log
and
T
log
.
Assertion
(iii)
follows
immediately
from
the
various
definitions
involved.
Q.E.D.
Definition
5.5.
In
the
notation
of
Definition
5.3,
Lemma
5.4,
we
shall
write
t
log
for
the
log
scheme
obtained
by
equipping
the
closed
point
t
of
T
with
the
log
structure
naturally
induced
by
the
log
structure
of
T
log
;
X
t
log
for
the
stable
log
curve
over
t
log
corresponding
to
the
natural
log
strict
morphism
t
log
(
→
T
log
)
→
(M
g,r
)
S
;
ρ
univ
:
I
T
log
−→
Dehn(G
X
log
)
X
log
t
for
the
homomorphism
obtained
in
Lemma
5.4,
(iii).
Proposition
5.6
(Outer
representations
arising
from
stable
log
curves).
In
the
notation
of
Definition
5.3,
Lemma
5.4,
the
following
hold:
(i)
(Compatibility
of
Σ-lengths)
Node(G
X
log
)
of
G
X
log
,
it
holds
that
For
each
node
e
∈
Σ
lng
Σ
G
log
(e,
ρ
X
s
log
)
=
lng
X
log
(e)
X
[cf.
Definitions
5.1;
5.3,
(ii)].
(ii)
(Isomorphicity
of
local
geometric
universal
outer
rep-
resentations)
The
homomorphism
:
I
T
log
−→
Dehn(G
X
log
)
ρ
univ
X
log
t
[cf.
Definition
5.5]
is
an
isomorphism
of
profinite
groups.
98
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(iii)
(Compatibility
with
generization)
Let
Q
⊆
Node(G
X
log
)
be
a
subset
of
Node(G
X
log
).
Then
there
exist
a
stable
log
curve
Y
log
over
S
log
and
an
isomorphism
of
semi-graphs
of
anabe-
∼
lioids
(G
X
log
)
Q
→
G
Y
log
that
fit
into
a
commutative
diagram
of
profinite
groups
ρ
univ
log
Y
I
T
log
−−−
t
−→
Dehn(G
Y
log
)
⏐
⏐
Y
⏐
⏐
ρ
univ
log
X
I
T
log
−−−
t
−→
Dehn(G
X
log
)
X
—
where
we
write
I
T
log
,
I
T
log
for
the
“I
T
log
”
associated
to
X
log
,
X
Y
Y
log
,
respectively;
the
right-hand
vertical
arrow
is
the
natural
∼
inclusion
induced,
via
the
isomorphism
(G
X
log
)
Q
→
G
Y
log
,
by
the
natural
inclusion
of
Theorem
4.8,
(ii);
the
left-hand
vertical
arrow
is
the
injection
induced,
via
the
[relevant]
isomorphism
of
Lemma
5.4,
(ii),
by
the
natural
projection
of
monoids
N
e
N
e
.
e∈Node(G
X
log
)
e∈Node(G
Y
log
)
[Note
that
it
follows
immediately
from
the
various
definitions
∼
involved
that
Node(G
Y
log
)
←
Node((G
X
log
)
Q
)
may
be
regarded
as
a
subset
of
Node(G
X
log
).]
(iv)
(Compatibility
with
specialization)
Let
H
be
a
semi-graph
∼
of
anabelioids
of
pro-Σ
PSC-type,
Q
⊆
Node(H),
and
H
Q
→
G
X
log
an
isomorphism
of
semi-graphs
of
anabelioids.
Then
there
exist
a
stable
log
curve
Y
log
over
S
log
and
an
isomor-
∼
phism
of
semi-graphs
of
anabelioids
H
→
G
Y
log
that
fit
into
a
commutative
diagram
of
profinite
groups
ρ
univ
log
X
I
T
log
−−−
t
−→
Dehn(G
X
log
)
⏐
X
⏐
⏐
⏐
ρ
univ
log
Y
I
T
log
−−−
t
−→
Dehn(G
Y
log
)
Y
—
where
we
write
I
T
log
,
I
T
log
for
the
“I
T
log
”
associated
to
X
log
,
X
Y
Y
log
,
respectively;
the
right-hand
vertical
arrow
is
the
natural
Combinatorial
anabelian
topics
I
99
∼
inclusion
induced,
via
the
isomorphisms
H
Q
→
G
X
log
and
∼
H
→
G
Y
log
,
by
the
natural
inclusion
of
Theorem
4.8,
(ii);
the
left-hand
vertical
arrow
is
the
injection
induced,
via
the
[rele-
vant]
isomorphism
of
Lemma
5.4,
(ii),
by
the
natural
projection
of
monoids
N
e
N
e
.
e∈Node(G
Y
log
)
e∈Node(G
X
log
)
[Note
that
it
follows
immediately
from
the
various
definitions
∼
involved
that
Node(G
X
log
)
←
Node(H
Q
)
may
be
regarded
as
∼
a
subset
of
Node(G
Y
log
)
←
Node(H).]
(v)
(Input
compatibility
with
“surgery”)
Let
H
be
a
sub-
semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
the
un-
derlying
semi-graph
of
G
X
log
,
Q
⊆
Node((G
X
log
)|
H
)
[cf.
Def-
inition
2.2,
(ii)]
a
subset
of
Node((G
X
log
)|
H
)
that
is
not
of
separating
type
[cf.
Definition
2.5,
(i)],
and
U
⊆
Cusp(((G
X
log
)|
H
)
Q
)
[cf.
Definition
2.5,
(ii)]
an
omittable
[cf.
Definition
2.4,
(i)]
subset
of
Cusp(((G
X
log
)|
H
)
Q
).
Then
there
exist
a
stable
log
curve
Y
log
over
S
log
and
an
isomor-
∼
phism
(((G
X
log
)|
H
)
Q
)
•U
→
G
Y
log
[cf.
Definition
2.4,
(ii)]
that
fit
into
a
commutative
diagram
of
profinite
groups
ρ
univ
log
X
I
S
log
−−−−→
I
T
log
−−−
t
−→
Dehn(G
X
log
)
⏐
X
⏐
⏐
⏐
ρ
univ
log
Y
I
S
log
−−−−→
I
T
log
−−−
t
−→
Dehn(G
Y
log
)
Y
—
where
we
write
I
T
log
,
I
T
log
for
the
“I
T
log
”
associated
to
X
Y
X
log
,
Y
log
,
respectively;
the
left-hand
horizontal
arrows
are
the
homomorphisms
induced
by
the
classifying
morphisms
associ-
ated
to
X
log
,
Y
log
,
respectively;
the
right-hand
vertical
arrow
is
the
natural
surjection
induced,
via
the
isomorphism
∼
(((G
X
log
)|
H
)
Q
)
•U
→
G
Y
log
,
by
the
natural
surjection
of
The-
orem
4.8,
(iii);
the
middle
vertical
arrow
is
the
surjection
in-
duced,
via
the
[relevant]
isomorphism
of
Lemma
5.4,
(ii),
by
the
natural
inclusion
of
monoids
N
e
→
N
e
.
e∈Node(G
Y
log
)
e∈Node(G
X
log
)
100
Yuichiro
Hoshi
and
Shinichi
Mochizuki
[Note
that
it
follows
immediately
from
the
various
definitions
∼
involved
that
Node(G
Y
log
)
←
Node((((G
X
log
)|
H
)
Q
)
•U
)
may
be
regarded
as
a
subset
of
Node(G
X
log
).]
(vi)
(Output
compatibility
with
“surgery”)
Let
H
be
a
semi-
graph
of
anabelioids
of
pro-Σ
PSC-type,
K
a
sub-semi-graph
of
PSC-type
[cf.
Definition
2.2,
(i)]
of
the
underlying
semi-
graph
of
H,
Q
⊆
Node(H|
K
)
[cf.
Definition
2.2,
(ii)]
a
subset
of
Node(H|
K
)
that
is
not
of
separating
type
[cf.
Defini-
tion
2.5,
(i)],
U
⊆
Cusp((H|
K
)
Q
)
[cf.
Definition
2.5,
(ii)]
an
omittable
[cf.
Definition
2.4,
(i)]
subset
of
Cusp((H|
K
)
Q
),
∼
and
((H|
K
)
Q
)
•U
→
G
X
log
[cf.
Definition
2.4,
(ii)]
an
isomor-
phism
of
semi-graphs
of
anabelioids.
Then
there
exist
a
stable
log
curve
Y
log
over
S
log
and
an
isomorphism
of
semi-graphs
of
∼
anabelioids
H
→
G
Y
log
that
fit
into
a
commutative
diagram
of
profinite
groups
ρ
univ
log
Y
I
S
log
−−−−→
I
T
log
−−−
t
−→
Dehn(G
Y
log
)
⏐
⏐
Y
⏐
⏐
ρ
univ
log
X
I
S
log
−−−−→
I
T
log
−−−
t
−→
Dehn(G
X
log
)
X
—
where
we
write
I
T
log
,
I
T
log
for
the
“I
T
log
”
associated
to
X
Y
X
log
,
Y
log
,
respectively;
the
left-hand
horizontal
arrows
are
the
homomorphisms
induced
by
the
classifying
morphisms
as-
sociated
to
Y
log
,
X
log
,
respectively;
the
right-hand
vertical
ar-
row
is
the
natural
surjection
induced,
via
the
isomorphisms
∼
∼
((H|
K
)
Q
)
•U
→
G
X
log
and
H
→
G
Y
log
,
by
the
natural
surjec-
tion
of
Theorem
4.8,
(iii);
the
middle
vertical
arrow
is
the
sur-
jection
induced,
via
the
[relevant]
isomorphism
of
Lemma
5.4,
(ii),
by
the
natural
inclusion
of
monoids
N
e
→
N
e
.
e∈Node(G
X
log
)
e∈Node(G
Y
log
)
[Note
that
it
follows
immediately
from
the
various
definitions
∼
involved
that
Node(G
X
log
)
←
Node(((H|
K
)
Q
)
•U
)
may
be
re-
∼
garded
as
a
subset
of
Node(G
Y
log
)
←
Node(H).]
Proof.
Assertion
(i)
follows
immediately
from
the
well-known
lo-
cal
structure
of
a
stable
log
curve
in
a
neighborhood
of
a
node.
Next,
Combinatorial
anabelian
topics
I
101
we
verify
assertion
(ii).
By
allowing
“ρ
X
s
log
”
to
vary
among
the
natu-
ral
outer
representations
I
S
log
→
Dehn(G
X
log
)
associated
to
stable
log
curves
“X
log
”
over
S
log
whose
classifying
morphisms
“σ”
coincide
with
the
given
σ
on
the
closed
point
s
of
S,
one
concludes
that
the
surjec-
follows
immediately
from
the
final
portion
of
Lemma
5.2,
tivity
of
ρ
univ
X
log
t
(ii),
concerning
ρ
of
SNN-type
[cf.
also
assertion
(i);
Theorem
4.8,
(iv)].
[Here,
we
recall
that
ρ
X
s
log
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)],
hence
also
of
SNN-type
[cf.
[NodNon],
Remark
2.4.2].]
On
the
Σ
-modules
of
other
hand,
since
both
Dehn(G
X
log
)
and
I
T
log
are
free
Z
rank
Node(G
X
log
)
[cf.
Theorem
4.8,
(iv);
Lemma
5.4,
(ii)],
assertion
(ii)
.
This
completes
the
follows
immediately
from
this
surjectivity
of
ρ
univ
X
log
t
proof
of
assertion
(ii).
Assertion
(iii)
(respectively,
(iv))
follows
immediately,
in
light
of
log
the
well-known
structure
of
(M
g,r
)
S
[cf.
also
the
discussion
entitled
“The
Étale
Fundamental
Group
of
a
Log
Scheme”
in
[CmbCsp],
§0,
concerning
the
specialization
isomorphism
on
fundamental
groups,
as
well
as
Remark
5.6.1
below],
by
considering
a
lifting
to
S
log
of
a
sta-
ble
log
curve
over
s
log
obtained
by
deforming
the
nodes
of
the
special
def
fiber
X
s
log
=
X
log
×
S
log
s
log
corresponding
to
the
nodes
contained
in
Q
(respectively,
degenerating
the
moduli
of
X
s
log
so
as
to
obtain
nodes
corresponding
to
the
nodes
contained
in
Q)
[cf.
also
Proposition
4.10,
(iii)].
Next,
we
verify
assertion
(v).
First,
we
observe
that
one
may
verify
easily
that
if
H
is
the
underlying
semi-graph
of
G
X
log
,
and
Q
=
∅,
then
the
stable
log
curve
Y
log
over
S
log
obtained
by
omitting
the
cusps
of
X
log
∼
contained
in
U
and
the
resulting
natural
isomorphism
(G
X
log
)
•U
→
G
Y
log
satisfy
the
conditions
given
in
the
statement
of
assertion
(v).
Thus,
one
verifies
immediately
that
to
verify
assertion
(v),
we
may
assume
without
loss
of
generality
that
U
=
∅.
def
def
Write
H
=
((G
X
log
)|
H
)
Q
and
V
=
Vert(G
X
log
)
\
Vert((G
X
log
)|
H
)
⊆
Vert(G
X
log
).
Denote
by
(g
H
,
r
H
)
the
type
of
H,
and,
for
each
v
∈
V
,
by
(g
v
,
r
v
)
the
type
of
v
[cf.
Definition
2.3,
(i),
(iii);
Remark
2.3.1].
Then
it
follows
immediately
from
the
general
theory
of
stable
log
curves
that
there
exists
a
“clutching
(1-)morphism”
corresponding
to
the
operations
“(−)|
H
”
and
“(−)
Q
”
[i.e.,
obtained
by
forming
appropriate
composites
of
the
clutching
morphisms
discussed
in
[Knud],
Definition
3.6]
def
N
=
(M
g
H
,r
H
)
s
×
s
v∈V
(M
g
v
,r
v
)
s
−→
(M
g,r
)
s
102
Yuichiro
Hoshi
and
Shinichi
Mochizuki
—
where
the
fiber
product
“
v∈V
”
is
taken
over
s
—
that
satisfies
the
following
condition:
write
N
log
for
the
log
stack
obtained
by
equip-
ping
the
stack
N
with
the
log
structure
induced
by
the
log
structure
log
of
(M
g,r
)
s
via
the
above
clutching
morphism;
then
there
exists
an
s
log
-
log
log
∈
N
log
(s
log
)
of
N
log
such
that
the
image
of
σ
N
via
valued
point
σ
N
log
the
natural
strict
(1-)morphism
N
log
→
(M
g,r
)
s
coincides
with
the
s
log
-
log
valued
point
of
(M
g,r
)
s
obtained
by
restricting
the
classifying
morphism
log
σ
log
∈
(M
g,r
)
S
(S
log
)
of
X
log
to
s
log
.
If,
moreover,
we
write
Y
s
log
for
the
log
∈
N
log
(s
log
)
stable
log
curve
over
s
log
corresponding
to
the
image
of
σ
N
via
the
composite
of
(1-)morphisms
def
log
N
log
−→
N
log
=
(M
g
H
,r
H
)
s
×
s
pr
log
log
1
(M
g
v
,r
v
)
s
−→
(M
g
H
,r
H
)
s
v∈V
—
where
the
first
arrow
is
the
(1-)morphism
of
log
stacks
obtained
by
“forgetting”
the
portion
of
the
log
structure
of
N
log
that
arises
from
[the
log
portion
of
the
log
structure
of
(M
g,r
)
s
determined
by]
the
irreducible
components
of
the
divisor
(M
g,r
)
s
\
(M
g,r
)
s
which
contain
the
image
of
N
→
(M
g,r
)
s
—
then
one
verifies
immediately
that,
for
any
stable
log
curve
Y
log
over
S
log
that
lifts
Y
s
log
,
there
exists
a
natural
identification
∼
isomorphism
H
=
((G
X
log
)|
H
)
Q
→
G
Y
log
.
Next,
let
us
observe
that
by
applying
the
various
definitions
in-
log
volved,
together
with
the
fact
that
the
(1-)morphism
N
log
→
(M
g,r
)
S
is
strict,
one
may
verify
easily
that
the
restrictions
of
the
natural
(1-
)morphisms
of
log
stacks
log
pr
log
log
1
N
log
←−
N
log
−→
(M
g,r
)
s
(M
g
H
,r
H
)
s
←−
to
a
suitable
étale
neighborhood
of
the
underlying
morphism
of
stacks
log
of
σ
N
∈
N
log
(s
log
)
induce
the
following
morphisms
between
the
charts
log
log
of
(M
g
H
,r
H
)
s
,
N
log
,
N
log
,
and
(M
g,r
)
s
determined
by
the
chart
of
log
“(M
g
•
,r
•
)
s
”
given
in
Lemma
5.4,
(i):
∼
N
→
N
⊕
{0}
e∈Node(H)
e
e∈Node(H)
e
→
e∈Node(G
)
X
log
N
e
∼
←
e∈Node(G
X
log
)
N
e
—
where
we
use
the
notation
N
e
to
denote
a
copy
of
the
monoid
N
indexed
by
e,
and
the
“
→”
is
the
natural
inclusion
determined
by
the
Combinatorial
anabelian
topics
I
103
natural
inclusion
Node(H)
→
Node(G).
Thus,
by
applying
the
func-
Σ
(1))”
to
the
homomorphism
tor
“Hom
Z
Σ
((−)
gp
,
Z
e∈Node(H)
N
e
→
N
obtained
by
composing
the
morphisms
of
the
above
e∈Node(G
)
e
X
log
display
and
considering
the
[relevant]
isomorphism
of
Lemma
5.4,
(ii),
we
obtain
a
homomorphism
I
T
log
→
I
T
log
,
which
makes
the
left-hand
X
Y
square
of
the
diagram
in
the
statement
of
assertion
(v)
commute.
On
the
other
hand,
to
verify
the
commutativity
of
the
right-hand
square
of
the
diagram
in
the
statement
of
assertion
(v),
let
us
observe
that
by
Theorem
4.8,
(iv),
it
suffices
to
verify
that
for
any
node
e
∈
Node(G
Y
log
)
of
G
Y
log
,
the
two
composites
ρ
univ
log
D
e
X
∼
X
t
Dehn(G
X
log
)
−→
Λ
G
log
−→
Λ
G
log
;
I
T
log
−→
X
X
ρ
univ
log
Y
Y
D
t
e
I
T
log
−→
I
T
log
−→
Dehn(G
Y
log
)
−→
Λ
G
log
X
Y
Y
—
where
we
write
e
X
for
the
node
of
G
X
log
corresponding
to
the
node
e
∈
Node(G
Y
log
)
via
the
natural
inclusion
Node(G
Y
log
)
→
Node(G
X
log
)
—
coincide.
But
this
follows
immediately
by
comparing
the
natural
action
of
I
T
log
on
the
portion
of
G
X
log
corresponding
to
{e
X
}
∪
V(e
X
)
X
with
the
natural
action
of
I
T
log
on
the
portion
of
G
Y
log
corresponding
to
Y
{e}
∪
V(e).
This
completes
the
proof
of
assertion
(v).
Finally,
we
verify
assertion
(vi).
First,
we
observe
that
one
may
verify
easily
that
if
K
is
the
underlying
semi-graph
of
H,
and
Q
=
∅,
then
the
stable
log
curve
Y
log
over
S
log
obtained
by
equipping
X
log
with
∼
suitable
cusps
satisfies,
for
a
suitable
choice
of
isomorphism
H
→
G
Y
log
,
the
conditions
given
in
the
statement
of
assertion
(vi).
Thus,
one
verifies
immediately
that
to
verify
assertion
(vi),
we
may
assume
without
loss
of
generality
that
U
=
∅.
def
Write
V
=
Vert(H)
\
Vert(H|
K
)
⊆
Vert(H).
Denote
by
(g
H
,
r
H
)
the
type
of
H,
and,
for
each
v
∈
V
,
by
(g
v
,
r
v
)
the
type
of
v.
Then
it
follows
immediately
from
the
general
theory
of
stable
log
curves
that
there
exists
a
clutching
“(1-)morphism”
corresponding
to
the
operations
“(−)|
K
”
and
“(−)
Q
”
[i.e.,
obtained
by
forming
appropriate
composites
of
the
clutching
morphisms
discussed
in
[Knud],
Definition
3.6]
def
(M
g
v
,r
v
)
s
−→
(M
g
H
,r
H
)
s
N
=
(M
g,r
)
s
×
s
v∈V
—
where
the
fiber
product
“
v∈V
”
is
taken
over
s
—
that
satisfies
the
following
condition:
write
N
log
for
the
log
stack
obtained
by
equip-
ping
the
stack
N
with
the
log
structure
induced
by
the
log
structure
104
Yuichiro
Hoshi
and
Shinichi
Mochizuki
log
of
(M
g
H
,r
H
)
s
via
the
above
clutching
morphism;
then
there
exists
an
log
log
∈
N
log
(s
log
)
of
N
log
such
that
the
image
of
σ
N
∈
s
log
-valued
point
σ
N
log
log
N
(s
)
via
the
composite
of
(1-)morphisms
def
log
N
log
−→
N
log
=
(M
g,r
)
s
×
s
pr
log
log
1
(M
g
v
,r
v
)
s
−→
(M
g,r
)
s
v∈V
—
where
the
first
arrow
is
the
(1-)morphism
of
log
stacks
obtained
by
“forgetting”
the
portion
of
the
log
structure
of
N
log
that
arises
from
log
[the
portion
of
the
log
structure
of
(M
g
H
,r
H
)
s
determined
by]
the
irre-
ducible
components
of
the
divisor
(M
g
H
,r
H
)
s
\
(M
g
H
,r
H
)
s
which
con-
tain
the
image
of
N
→
(M
g
H
,r
H
)
s
—
coincides
with
the
s
log
-valued
log
point
of
(M
g,r
)
s
obtained
by
restricting
the
classifying
morphism
σ
log
∈
log
(M
g,r
)
S
(S
log
)
of
X
log
to
s
log
.
If,
moreover,
we
write
Y
s
log
for
the
stable
log
log
curve
over
s
log
corresponding
to
the
image
of
σ
N
∈
N
log
(s
log
)
via
log
the
natural
strict
(1-)morphism
N
log
→
(M
g
H
,r
H
)
s
,
then
one
verifies
immediately
that,
for
any
stable
log
curve
Y
log
over
S
log
that
lifts
Y
s
log
,
there
exist
a
sub-semi-graph
of
PSC-type
K
of
the
underlying
semi-graph
of
G
Y
log
,
a
subset
Q
⊆
Node((G
Y
log
)|
K
),
and
an
isomorphism
of
semi-
∼
graphs
of
anabelioids
H
→
G
Y
log
that
satisfy
the
following
conditions:
(a)
((G
Y
log
)|
K
)
Q
may
be
naturally
identified
with
G
X
log
.
∼
∼
(b)
The
isomorphism
H
→
G
Y
log
induces
an
isomorphism
K
→
K
∼
∼
and
a
bijection
Q
→
Q
,
hence
also
an
isomorphism
(H|
K
)
Q
→
((G
Y
log
)|
K
)
Q
.
(c)
The
automorphism
of
G
X
log
determined
by
the
composite
∼
∼
∼
G
X
log
←−
(H|
K
)
Q
−→
((G
Y
log
)|
K
)
Q
−→
G
X
log
—
where
the
first
arrow
is
the
isomorphism
given
in
the
state-
ment
of
assertion
(vi);
the
second
arrow
is
the
isomorphism
of
(b);
the
third
arrow
is
the
natural
isomorphism
arising
from
the
natural
identification
of
(a)
—
is
contained
in
Aut
|grph|
(G
X
log
),
and,
moreover,
the
automorphism
of
Λ
G
log
induced
by
this
au-
X
tomorphism
of
G
X
log
is
the
identity
automorphism
[cf.
Propo-
sition
4.10,
(iii)].
Thus,
by
applying
a
similar
argument
to
the
argument
used
in
the
proof
of
assertion
(v),
one
verifies
easily
that
the
stable
log
curve
Y
log
and
the
Combinatorial
anabelian
topics
I
105
∼
isomorphism
H
→
G
Y
log
satisfy
the
conditions
given
in
the
statement
of
assertion
(vi).
This
completes
the
proof
of
assertion
(vi).
Q.E.D.
Remark
5.6.1.
Here,
we
take
the
opportunity
to
correct
a
minor
misprint
in
the
discussion
entitled
“The
Étale
Fundamental
Group
of
a
Log
Scheme”
in
[CmbCsp],
§0.
In
the
third
paragraph
of
this
discussion,
the
field
K
should
be
defined
as
a
maximal
algebraic
extension
of
K
◦
among
those
extensions
which
are
unramified
over
U
S
◦
[i.e.,
but
not
necessarily
over
R
◦
].
Theorem
5.7
(Compatibility
of
scheme-theoretic
and
ab-
stract
combinatorial
cyclotomic
synchronizations).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g−2+r
>
0;
Σ
a
nonempty
set
of
prime
numbers;
R
a
complete
discrete
valuation
ring
whose
residue
field
is
separably
closed
of
characteristic
∈
Σ;
S
log
the
log
scheme
obtained
by
def
equipping
S
=
Spec
R
with
the
log
structure
defined
by
its
closed
point;
log
a
stable
log
curve
of
type
(g,
r)
over
S
log
;
G
X
log
the
semi-graph
X
of
anabelioids
of
pro-Σ
PSC-type
determined
by
the
special
fiber
of
the
the
completion
of
stable
log
curve
X
log
[cf.
[CmbGC],
Example
2.5];
O
the
local
ring
of
(M
g,r
)
S
[cf.
the
discussion
entitled
“Curves”
in
§0]
at
the
image
of
the
closed
point
of
S
via
the
underlying
(1-)morphism
of
stacks
σ
:
S
→
(M
g,r
)
S
of
the
classifying
morphism
of
X
log
;
T
log
for
def
with
the
log
struc-
the
log
scheme
obtained
by
equipping
T
=
Spec
O
log
ture
induced
by
the
log
structure
of
(M
g,r
)
S
[cf.
the
discussion
entitled
“Curves”
in
§0];
I
T
log
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
π
1
(T
log
)
of
T
log
.
Then
there
exists
an
isomorphism
∼
Σ
(1))
−→
Λ
G
syn
X
log
:
Λ
Σ
=
Hom(N
gp
,
Z
log
def
X
[cf.
Definition
3.8,
(i)]
such
that
the
composite
∼
Σ
I
T
log
→
e∈Node(G
)
Λ
[e]
X
log
syn
X
log
∼
→
e∈Node(G
X
log
)
Λ
G
X
log
D
G
X
log
∼
←
Dehn(G
X
log
)
[cf.
Definitions
4.4;
4.7]
—
where
we
use
the
notation
Λ
Σ
[e]
to
denote
a
copy
of
Λ
Σ
indexed
by
e
∈
Node(G
X
log
),
and
the
first
arrow
is
the
[relevant]
isomorphism
of
Lemma
5.4,
(ii)
—
coincides
with
the
outer
:
I
T
log
→
Dehn(G
X
log
)
[cf.
Definition
5.5]
associated
representation
ρ
univ
X
log
t
106
Yuichiro
Hoshi
and
Shinichi
Mochizuki
to
the
stable
log
curve
over
T
log
corresponding
to
the
tautological
strict
log
(1-)morphism
T
log
→
(M
g,r
)
S
.
Proof.
In
light
of
Theorem
4.8,
(ii),
(iv);
Proposition
5.6,
(ii),
by
applying
Proposition
5.6,
(iii),
to
the
various
generizations
of
the
form
“(G
X
log
)
Node(G
log
)\{e}
”,
it
follows
immediately
that
for
each
node
e
∈
X
Node(G
X
log
),
there
exists
a(n)
[necessarily
unique]
isomorphism
∼
syn
X
log
[e]
:
Λ
Σ
[e]
−→
Λ
G
log
X
—
where
Λ
Σ
[e]
is
a
copy
of
Λ
Σ
indexed
by
e
∈
Node(G
X
log
)
—
such
that
the
composite
∼
Σ
→
I
T
log
e∈Node(G
)
Λ
[e]
X
log
e
syn
X
log
[e]
∼
→
e∈Node(G
X
log
D
G
X
log
∼
←
)
Λ
G
X
log
Dehn(G
X
log
)
∼
—
where
the
first
“
→”
is
the
[relevant]
isomorphism
of
Lemma
5.4,
(ii)
.
—
coincides
with
ρ
univ
X
t
log
Thus,
to
complete
the
proof
of
Theorem
5.7,
it
suffices
to
verify
that
this
isomorphism
syn
X
log
[e]
is
independent
of
the
choice
of
e.
Now
if
Node(G
X
log
)
≤
1,
then
this
independence
is
immediate.
Thus,
suppose
that
Node(G
X
log
)
>
1
and
fix
two
distinct
nodes
e
1
,
e
2
∈
Node(G
X
log
)
of
G
X
log
.
The
rest
of
the
proof
of
Theorem
5.7
is
devoted
to
verifying
that
(‡):
the
two
isomorphisms
Σ
Λ
[e
1
]
syn
X
log
[e
1
]
∼
−→
Σ
Λ
G
log
,
Λ
[e
2
]
X
syn
X
log
[e
2
]
∼
−→
Λ
G
log
X
coincide.
Next,
let
us
observe
that
one
may
verify
easily
that
there
exist
•
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
H
∗
,
•
a
sub-semi-graph
of
PSC-type
K
∗
of
the
underlying
semi-graph
of
H
∗
,
•
an
omittable
subset
Q
∗
⊆
Cusp((H
∗
)|
K
∗
),
and
•
an
isomorphism
∼
((H
∗
)|
K
∗
)
•Q
∗
−→
G
X
log
Combinatorial
anabelian
topics
I
107
such
that
the
subset
U
∗
⊆
Node(H
∗
)
corresponding,
relative
to
the
iso-
∼
morphism
((H
∗
)|
K
∗
)
•Q
∗
→
G
X
log
,
to
the
subset
{e
1
,
e
2
}
⊆
Node(G
X
log
)
is
not
of
separating
type.
Thus,
it
follows
immediately
from
Propo-
sition
5.6,
(vi)
—
i.e.,
by
replacing
X
log
(respectively,
e
1
,
e
2
)
by
the
stable
log
curve
“Y
log
”
obtained
by
applying
Proposition
5.6,
(vi),
to
∼
the
isomorphism
((H
∗
)|
K
∗
)
•Q
∗
→
G
X
log
(respectively,
by
the
two
nodes
∈
Node(G
Y
log
)
corresponding
to
the
two
nodes
∈
U
∗
)
—
that
to
verify
the
above
(‡),
we
may
assume
without
loss
of
generality
that
the
subset
{e
1
,
e
2
}
⊆
Node(G
X
log
)
is
not
of
separating
type.
Thus,
it
follows
immediately
from
Proposition
5.6,
(iii)
—
i.e.,
by
replacing
X
log
(respectively,
e
1
,
e
2
)
by
the
stable
log
curve
“Y
log
”
ob-
tained
by
applying
Proposition
5.6,
(iii),
to
(G
X
log
)
Node(G
log
)\{e
1
,e
2
}
X
(respectively,
by
the
two
nodes
∈
Node(G
Y
log
)
corresponding
to
e
1
,
e
2
)
—
that
to
verify
the
above
(‡),
we
may
assume
without
loss
of
general-
ity
that
Node(G
X
log
)
=
{e
1
,
e
2
},
and
that
Node(G
X
log
)
=
{e
1
,
e
2
}
is
not
of
separating
type.
One
verifies
easily
that
these
hypotheses
imply
that
Vert(G
X
log
)
=
1.
Next,
let
us
observe
that
one
may
verify
easily
that
there
exist
[cf.
Fig.
6
below]
•
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
H
†
,
•
two
distinct
cusps
c
†
1
,
c
†
2
∈
Cusp(H
†
)
of
H
†
,
•
three
distinct
nodes
f
1
†
,
f
2
†
,
f
3
†
∈
Node(H
†
)
of
H
†
,
and
•
an
isomorphism
(H
†
)
†
†
{f
1
†
,f
2
†
,f
3
†
}
•{c
1
,c
2
}
∼
−→
G
X
log
such
that
•
Vert(H
†
)
=
{v
1
†
,
v
2
†
,
v
3
†
,
v
4
†
};
•
for
i
∈
{1,
2},
if
we
write
e
†
i
∈
Node(H
†
)
for
the
node
corre-
∼
sponding,
relative
to
the
isomorphism
(H
†
†
†
†
)
•{c
†
,c
†
}
→
{f
1
,f
2
,f
3
}
1
2
G
X
log
,
to
e
i
∈
Node(G
X
log
),
then
it
holds
that
V(e
†
i
)
=
{v
i
†
};
•
V(f
1
†
)
=
{v
1
†
,
v
3
†
},
V(f
2
†
)
=
{v
2
†
,
v
3
†
},
V(f
3
†
)
=
{v
3
†
,
v
4
†
};
•
V(c
†
1
)
=
V(c
†
2
)
=
{v
4
†
};
•
for
i
∈
{1,
2,
3},
v
i
†
is
of
type
(0,
3)
[cf.
Definition
2.3,
(iii)].
108
Yuichiro
Hoshi
and
Shinichi
Mochizuki
◦
◦
·
·
··
·
·
•
v
4
†
f
3
†
v
1
†
e
†
1
•
v
2
†
f
1
†
•
v
3
†
f
2
†
•
e
†
2
Figure
6:
The
underlying
semi-graph
of
H
†
One
verifies
easily
that
these
hypotheses
imply
that
(N
(v
1
†
)
∩
N
(v
3
†
))
=
(N
(v
2
†
)
∩
N
(v
3
†
))
=
1.
Thus,
it
follows
immediately
from
Proposition
5.6,
(iv),
(vi)
—
i.e.,
by
replacing
X
log
(respectively,
e
1
,
e
2
)
by
the
stable
log
curve
“Y
log
”
obtained
by
applying
Proposition
5.6,
(iv),
∼
(vi),
to
the
isomorphism
(H
†
†
†
†
)
•{c
†
,c
†
}
→
G
X
log
(respectively,
by
{f
1
,f
2
,f
3
}
1
2
the
two
nodes
∈
Node(G
Y
log
)
corresponding
to
the
two
nodes
e
†
1
,
e
†
2
)
—
that
to
verify
the
above
(‡),
we
may
assume
without
loss
of
generality
that
there
exist
vertices
v
1
,
v
2
,
v
3
of
G
X
log
such
that
•
for
i
∈
{1,
2},
V(e
i
)
=
{v
i
};
•
for
i
∈
{1,
2,
3},
v
i
is
of
type
(0,
3);
•
(N
(v
1
)
∩
N
(v
3
))
=
(N
(v
2
)
∩
N
(v
3
))
=
1.
Write
H
for
the
sub-semi-graph
of
PSC-type
of
the
underlying
semi-
graph
of
G
X
log
whose
set
of
vertices
=
{v
1
,
v
2
,
v
3
}.
Then
one
verifies
eas-
ily
that
these
hypotheses
imply
that
Node((G
X
log
)|
H
)
=
{e
1
,
e
2
,
f
1
,
f
2
},
where
we
write
{f
1
}
=
N
(v
1
)
∩
N
(v
3
),
{f
2
}
=
N
(v
2
)
∩
N
(v
3
).
Thus,
it
follows
immediately
from
Proposition
5.6,
(v)
—
i.e.,
by
replacing
X
log
(respectively,
e
1
,
e
2
)
by
the
stable
log
curve
“Y
log
”
ob-
tained
by
applying
Proposition
5.6,
(v),
to
(G
X
log
)|
H
(respectively,
by
the
two
nodes
∈
Node(G
Y
log
)
corresponding
to
e
1
,
e
2
)
—
that
to
verify
the
above
(‡),
we
may
assume
without
loss
of
generality
that
there
exist
three
distinct
vertices
v
1
,
v
2
,
v
3
of
G
X
log
such
that
•
for
i
∈
{1,
2},
V(e
i
)
=
{v
i
};
Combinatorial
anabelian
topics
I
109
•
for
i
∈
{1,
2,
3},
v
i
is
of
type
(0,
3);
•
Node(G
X
log
)
=
{e
1
,
e
2
,
f
1
,
f
2
},
where
we
write
{f
1
}
=
(N
(v
1
)∩
N
(v
3
)),
{f
2
}
=
(N
(v
2
)
∩
N
(v
3
)).
One
verifies
easily
that
these
hypotheses
imply
that
there
exists
a
cusp
c
of
G
X
log
such
that
Cusp(G
X
log
)
=
{c}
=
C(v
3
).
Then
it
follows
immediately
from
the
explicit
structure
of
G
X
log
that
there
exists
an
automorphism
τ
of
X
t
log
[cf.
Definition
5.5]
such
that
the
∼
automorphism
of
Node(G
X
log
)
=
{e
1
,
e
2
,
f
1
,
f
2
}
(respectively,
I
T
log
→
Σ
(1))
[cf.
Lemma
5.4,
(ii)])
induced
by
Hom
Z
Σ
((N
e
1
⊕N
e
2
⊕N
f
1
⊕N
f
2
)
gp
,
Z
τ
is
given
by
mapping
e
1
→
e
2
,
e
2
→
e
1
,
f
1
→
f
2
,
f
2
→
f
1
,
(respectively,
by
the
corresponding
permutation
of
factors
of
N
e
1
⊕
N
e
2
⊕
N
f
1
⊕
N
f
2
),
and,
moreover,
τ
preserves
the
cusp
corresponding
to
c.
Now
it
follows
immediately
from
Corollary
3.9,
(v),
together
with
the
fact
that
the
automorphism
of
the
anabelioid
(G
X
log
)
c
corresponding
to
the
cusp
c
induced
by
τ
is
the
identity
automorphism
[cf.
the
argument
used
in
the
final
portion
of
the
proof
of
Corollary
3.9,
(vi)],
that
the
automorphism
of
Λ
G
log
induced
by
τ
is
the
identity
automorphism.
Thus,
by
applying
X
the
evident
functoriality
of
the
homomorphism
ρ
univ
with
respect
to
the
X
t
log
automorphism
of
G
X
log
induced
by
τ
,
one
concludes
immediately
from
the
above
description
of
τ
,
together
with
Theorem
4.8,
(v),
that
the
assertion
(‡)
holds.
This
completes
the
proof
of
Theorem
5.7.
Q.E.D.
Definition
5.8.
Let
α
∈
Dehn(G)
be
a
profinite
Dehn
multi-twist
of
G
and
u
∈
Λ
G
a
topological
generator
of
Λ
G
.
(i)
Σ
-
Let
e
∈
Node(G)
be
a
node
of
G.
Then
since
Λ
G
is
a
free
Z
module
of
rank
1
[cf.
Definition
3.8,
(i)],
there
exists
a
unique
Σ
of
Z
Σ
such
that
D
e
(α)
=
a
e
u.
We
shall
refer
element
a
e
∈
Z
Σ
as
the
Dehn
coordinate
of
α
indexed
by
e
with
to
a
e
∈
Z
respect
to
u.
(ii)
We
shall
say
that
a
profinite
Dehn
multi-twist
α
∈
Dehn(G)
is
nondegenerate
if,
for
each
node
e
∈
Node(G)
of
G,
the
Dehn
coordinate
of
α
indexed
by
e
with
respect
to
u
[cf.
(i)]
topo-
Σ
.
Note
that
it
is
logically
generates
an
open
subgroup
of
Z
immediate
that
if
α
is
nondegenerate,
then
the
Dehn
coordi-
∼
Σ
→
nate
(∈
Z
l∈Σ
Z
l
⊆
l∈Σ
Q
l
)
of
α
indexed
by
e
with
respect
to
u
is
contained
in
l∈Σ
Q
∗
l
.
110
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(iii)
We
shall
say
that
a
profinite
Dehn
multi-twist
α
∈
Dehn(G)
is
positive
definite
if
α
is
nondegenerate
[cf.
(ii)],
and,
moreover,
the
following
condition
is
satisfied:
For
each
node
e
∈
Node(G)
Σ
the
Dehn
coordinate
of
α
indexed
by
of
G,
denote
by
a
e
∈
Z
e
with
respect
to
u
[cf.
(i)].
[Thus,
a
e
∈
l∈Σ
Q
∗
l
—
cf.
(ii).]
Then
for
any
e,
e
∈
Node(G),
a
e
/a
e
is
contained
in
the
image
def
of
the
diagonal
map
Q
>0
=
{
a
∈
Q
|
a
>
0
}
→
l∈Σ
Q
∗
l
.
Remark
5.8.1.
One
may
verify
easily
that
the
notions
defined
in
Definition
5.8,
(ii),
(iii),
are
independent
of
the
choice
of
the
topological
generator
u
of
Λ
G
.
Corollary
5.9
(Properties
of
outer
representations
of
PSC–
type
and
profinite
Dehn
multi-twists).
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
ρ
:
I
→
Aut(G)
an
outer
representation
of
pro-Σ
PSC-type
[cf.
[NodNon],
Definition
2.1,
(i)].
Suppose
that
I
is
isomor-
Σ
.
Then
the
following
hold:
phic
to
Z
(i)
(ii)
(Outer
representations
of
SVA-type
and
profinite
Dehn
multi-twists)
The
following
three
conditions
are
equivalent:
(i-1)
ρ
is
of
SVA-type
[cf.
[NodNon],
Definition
2.4,
(ii)].
(i-2)
The
image
of
any
topological
generator
of
I
is
a
profinite
Dehn
multi-twist
[cf.
Definition
4.4].
(i-3)
There
exists
a
topological
generator
of
I
whose
image
via
ρ
is
a
profinite
Dehn
multi-twist.
(Outer
representations
of
SNN-type
and
nondegener-
ate
profinite
Dehn
multi-twists)
The
following
three
con-
ditions
are
equivalent
[cf.
the
related
discussion
of
[NodNon],
Remark
2.14.1]:
(ii-1)
ρ
is
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)].
(ii-2)
The
image
of
any
topological
generator
of
I
is
a
non-
degenerate
[cf.
Definition
5.8,
(ii)]
profinite
Dehn
multi-twist.
(ii-3)
There
exists
a
topological
generator
of
I
whose
image
via
ρ
is
a
nondegenerate
profinite
Dehn
multi-twist.
Combinatorial
anabelian
topics
I
(iii)
111
(Outer
representations
of
IPSC-type
and
positive
defi-
nite
profinite
Dehn
multi-twists)
The
following
three
con-
ditions
are
equivalent
[cf.
Remark
5.10.1
below;
the
related
discussion
of
[NodNon],
Remark
2.14.1]:
(iii-1)
ρ
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)].
(iii-2)
The
image
of
any
topological
generator
of
I
is
a
posi-
tive
definite
[cf.
Definition
5.8,
(iii)]
profinite
Dehn
multi-twist.
(iii-3)
There
exists
a
topological
generator
of
I
whose
image
via
ρ
is
a
positive
definite
profinite
Dehn
multi-twist.
(iv)
(Synchronization
associated
to
outer
representations
of
IPSC-type)
Suppose
that
ρ
is
of
IPSC-type.
Write
Σ
)
∗
Σ
)
+
⊆
(
Z
(
Z
def
for
the
intersection
of
the
images
of
the
diagonal
map
Q
>0
=
{
a
∈
Q
|
a
>
0
}
→
l∈Σ
Q
l
and
the
composite
of
natural
∼
Σ
)
∗
→
Z
Σ
→
morphisms
(
Z
l∈Σ
Z
l
⊆
l∈Σ
Q
l
.
[Thus,
when
Σ
+
Σ
=
Primes,
it
holds
that
(
Z
)
=
{1}.]
Then
there
exists
a
Σ
-modules
Σ
)
+
-orbit
of
isomorphisms
of
Z
natural
(
Z
∼
syn
ρ
:
I
−→
Λ
G
that
is
functorial,
in
ρ,
with
respect
to
isomorphisms
of
outer
representations
of
PSC-type
[cf.
[NodNon],
Definition
2.1,
(ii)].
(v)
(Compatibility
of
synchronizations
with
finite
étale
def
out
coverings)
In
the
situation
of
(iv),
let
Π
⊆
Π
I
=
Π
G
I
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
be
an
open
subgroup
of
Π
I
such
that
if
we
write
H
→
G
for
the
connected
finite
étale
covering
of
G
corresponding
to
Π
∩
Π
G
[so
Π
H
=
def
Π
∩
Π
G
],
then
the
outer
representation
ρ
Π
:
J
=
Π/Π
H
→
Σ
-modules
Out(Π
H
)
is
of
IPSC-type.
Then
the
diagram
of
Z
syn
ρ
Π
J
−−−−→
Λ
H
⏐
⏐
⏐
⏐
syn
ρ
I
−−−−→
Λ
G
112
Yuichiro
Hoshi
and
Shinichi
Mochizuki
—
where
the
left-hand
vertical
arrow
is
the
natural
inclusion;
the
right-hand
vertical
arrow
is
the
isomorphism
of
Corol-
lary
3.9,
(iii)
—
commutes
up
to
multiplication
by
an
el-
ement
∈
Q
>0
.
Proof.
Assertion
(i)
follows
immediately
from
condition
(2
)
of
[NodNon],
Definition
2.4.
Next,
we
verify
assertions
(ii)
and
(iii).
The
implication
(ii-1)
=⇒
(ii-2)
,
(respectively,
(iii-1)
=⇒
(iii-2))
follows
immediately
from
the
final
portion
of
Lemma
5.2,
(ii),
concern-
ing
ρ
of
SNN-type
(respectively,
Lemma
5.4,
(ii);
Theorem
5.7).
The
implications
(ii-2)
=⇒
(ii-3)
,
(iii-2)
=⇒
(iii-3)
are
immediate.
Next,
we
verify
the
implication
(ii-3)
=⇒
(ii-1)
.
It
follows
from
the
implication
(i-3)
⇒
(i-1)
that
ρ
is
of
SVA-type.
Thus,
to
show
the
implication
in
question,
it
suffices
to
verify
that
ρ
satisfies
be
an
ele-
condition
(3)
of
[NodNon],
Definition
2.4.
Let
e
∈
Node(
G)
out
Π
I
def
=
Π
G
I
[cf.
the
discussion
entitled
“Topological
ment
of
Node(
G);
the
two
distinct
elements
of
Vert(
G)
such
groups”
in
§0];
v
,
w
∈
Vert(
G)
that
V(
e
)
=
{
v
,
w}
[cf.
[NodNon],
Remark
1.2.1,
(iii)];
I
e
,
I
v
,
I
w
⊆
Π
I
the
inertia
subgroups
of
Π
I
associated
to
e
,
v
,
w,
respectively.
Then
since
the
homomorphisms
of
the
final
two
displays
of
Lemma
5.2,
(ii),
Σ
-modules
of
rank
1
[cf.
Defi-
coincide,
and
Λ
G
log
and
I
v
are
free
Z
X
nition
3.8,
(i);
[NodNon],
Lemma
2.5,
(i)],
it
follows
immediately
from
the
definition
of
nondegeneracy
that
the
composite
of
the
second
display
of
Lemma
5.2,
(ii),
is
an
open
injection.
Thus,
it
follows
immediately
that
the
natural
homomorphism
I
v
×
I
w
→
I
e
has
open
image,
and
that
I
v
∩
I
w
=
{1},
i.e.,
that
I
v
×
I
w
→
I
e
is
injective.
That
is
to
say,
ρ
satisfies
condition
(3)
of
[NodNon],
Definition
2.4.
This
completes
the
proof
of
the
implication
in
question.
Next,
we
verify
the
implication
(iii-3)
=⇒
(iii-1)
.
Let
u
∈
Λ
G
be
a
topological
generator
of
Λ
G
.
Then
it
follows
immediately
from
Lemma
5.4,
(i),
(ii),
and
Theorem
5.7
—
by
considering
the
stable
Combinatorial
anabelian
topics
I
113
log
curve
over
S
log
corresponding
to
a
suitable
homomorphism
of
R-
≃
R[[t
1
,
·
·
·
,
t
3g−g+r
]]
→
R
[cf.
Lemma
5.4,
(i)]
—
that
to
algebras
O
complete
the
proof
of
the
implication
in
question,
it
suffices
to
verify
that
there
exists
a
topological
generator
α
∈
I
of
I
which
satisfies
the
following
condition
(∗):
(∗):
The
Dehn
coordinates
of
ρ(α)
with
respect
to
u
def
[cf.
Definition
5.8,
(i)]
∈
N
=0
=
N
\
{0}.
To
this
end,
let
α
∈
I
be
a
topological
generator
of
I
such
that
ρ(α)
is
a
positive
definite
profinite
Dehn
multi-twist
of
G
[cf.
condition
(iii-3)].
Σ
the
Dehn
coordinate
For
each
node
e
∈
Node(G)
of
G,
denote
by
a
e
∈
Z
of
ρ(α)
indexed
by
e
with
respect
to
u.
Now
since
ρ(α)
is
nondegenerate,
it
follows
immediately
from
the
definition
of
nondegeneracy
that
for
each
Σ
)
∗
.
Thus,
it
follows
node
e
∈
Node(G)
of
G,
it
holds
that
a
e
∈
N
=0
·
(
Z
immediately
that
for
a
given
node
f
∈
Node(G)
of
G,
by
replacing
α
by
a
suitable
topological
generator
of
I,
we
may
assume
without
loss
of
generality
that
a
f
∈
N
=0
.
In
particular,
it
follows
immediately
from
the
definition
of
positive
definiteness
that
there
exists
an
element
a
∈
N
=0
such
that
for
each
node
e
∈
Node(G)
of
G,
it
holds
that
a
·
a
e
∈
N
=0
.
Moreover,
again
by
replacing
α
by
a
suitable
topological
generator
of
I,
we
may
assume
that
every
prime
number
dividing
a
belongs
to
Σ.
Σ
∩
(
1
·
N
=0
)
that
a
e
is
But
then
it
follows
from
the
fact
that
a
e
∈
Z
a
a
positive
rational
number
that
is
integral
at
every
element
of
Primes,
i.e.,
that
a
e
∈
N
=0
,
as
desired.
In
particular,
the
topological
generator
α
∈
I
of
I
satisfies
the
above
condition
(∗).
This
completes
the
proof
of
the
implication
in
question,
hence
also
of
assertions
(ii)
and
(iii).
Next,
we
verify
assertion
(iv).
It
follows
immediately
from
the
fi-
nal
portion
of
Lemma
5.2,
(ii),
concerning
ρ
of
SNN-type
that
for
each
e
∈
Node(G),
the
homomorphism
syn
ρ
:
I
→
Λ
G
obtained
by
dividing
ρ
D
the
composite
I
→
Dehn(G)
→
e
Λ
G
by
lng
Σ
G
(e,
ρ)
is
an
isomorphism.
Moreover,
by
“translating
into
group
theory”
the
scheme-theoretic
con-
tent
of
Lemma
5.4,
(ii),
by
means
of
the
correspondence
between
group-
theoretic
and
scheme-theoretic
notions
given
in
Proposition
5.6,
(i);
The-
orem
5.7,
one
concludes
that
syn
ρ
is
independent
—
up
to
multiplication
Σ
)
+
—
of
the
choice
of
the
node
e
∈
Node(G).
Now
by
an
element
of
(
Z
the
functoriality
of
syn
ρ
follows
immediately
from
the
functoriality
of
the
homomorphism
D
e
[cf.
Theorem
4.8,
(iv)],
together
with
the
group-
theoreticity
of
lng
Σ
G
(e,
ρ).
This
completes
the
proof
of
assertion
(iv).
Finally,
assertion
(v)
follows
immediately,
in
light
of
the
group-
theoretic
construction
of
“syn
ρ
”
given
in
the
proof
of
assertion
(iv),
from
the
various
definitions
involved.
Q.E.D.
114
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Remark
5.9.1.
(i)
Corollary
5.9,
(iv),
may
be
regarded
as
a
sort
of
abstract
com-
binatorial
analogue
of
the
cyclotomic
synchronization
given
in
[GalSct],
Theorem
4.3
[cf.
also
[AbsHyp],
Lemma
2.5,
(ii)].
(ii)
It
follows
from
Theorem
5.7
that
one
may
think
of
the
isomor-
phisms
of
Corollary
5.9,
(iv),
as
a
sort
of
abstract
combinato-
rial
construction
of
the
various
identification
isomorphisms
be-
Σ
(1)”
that
appear
in
Lemma
5.4,
tween
the
various
copies
of
“
Z
(ii).
Such
identification
isomorphisms
are
typically
“taken
for
granted”
in
conventional
discussions
of
scheme
theory.
Remark
5.9.2.
(i)
Σ
-modules
Consider
the
exact
sequence
of
free
Z
def
def
comb
0
−→
M
G
vert
−→
M
G
=
Π
ab
=
M
G
/M
G
vert
−→
0
G
−→
M
G
Σ
-submodule
of
M
G
—
where
we
write
M
G
vert
⊆
M
G
for
the
Z
topologically
generated
by
the
images
of
the
verticial
subgroups
of
Π
G
[cf.
[CmbGC],
Remark
1.1.4].
Then
one
verifies
easily
that
any
profinite
Dehn
multi-twist
α
∈
Dehn(G)
preserves
and
induces
the
identity
automorphism
on
M
G
vert
,
M
G
comb
.
In
par-
ticular,
the
homomorphism
M
G
→
M
G
obtained
by
considering
the
difference
of
the
automorphism
of
M
G
induced
by
α
and
the
identity
automorphism
on
M
G
naturally
determines
[and
is
determined
by!]
a
homomorphism
α
comb,vert
:
M
G
comb
−→
M
G
vert
.
Σ
-submodule
topologically
gen-
Write
M
G
edge
⊆
M
G
vert
for
the
Z
erated
by
the
image
of
the
edge-like
subgroups
of
Π
G
.
Then
the
following
two
facts
are
well-known:
∼
•
If
Cusp(G)
=
∅,
then
Poincaré
duality
M
G
→
∼
Hom
Z
Σ
(M
G
,
Λ
G
)
determines
an
isomorphism
M
G
edge
→
Hom
Z
Σ
(M
G
comb
,
Λ
G
)
[cf.
[CmbGC],
Proposition
1.3].
•
The
natural
homomorphism
Dehn(G)
−→
Hom
Z
Σ
(M
G
comb
,
M
G
vert
)
given
by
mapping
α
→
α
comb,vert
factors
through
the
sub-
module
Hom
Z
Σ
(M
G
comb
,
M
G
edge
)
⊆
Hom
Z
Σ
(M
G
comb
,
M
G
vert
).
Combinatorial
anabelian
topics
I
115
[Indeed,
this
may
be
verified,
for
instance,
by
applying
a
similar
argument
to
the
argument
used
in
the
proof
of
[CmbGC],
Proposition
1.3,
involving
weights.]
Thus,
if
Cusp(G)
=
∅,
then
we
obtain
a
homomorphism
Σ
)
Ω
G
:
Dehn(G)
−→
M
G
edge
⊗
Z
Σ
M
G
edge
⊗
Z
Σ
Hom
Z
Σ
(Λ
G
,
Z
that
is
manifestly
functorial,
in
G,
with
respect
to
isomor-
phisms
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type.
The
matrices
that
appear
in
the
image
of
this
homomorphism
Ω
G
are
often
referred
to
as
period
matrices.
(ii)
Now
let
us
recall
that
[CmbGC],
Proposition
2.6,
plays
a
key
role
in
the
proof
of
the
combinatorial
version
of
the
Grothendieck
conjecture
given
in
[CmbGC],
Corollary
2.7,
(iii).
Moreover,
the
proof
of
[CmbGC],
Proposition
2.6,
is
essentially
a
formal
consequence
of
the
nondegeneracy
of
the
period
matrix
associated
to
a
positive
definite
profinite
Dehn
multi-twist
—
i.e.,
of
the
injectivity
of
the
homomorphism
α
comb,vert
:
M
G
comb
−→
M
G
vert
of
(i)
in
the
case
where
α
∈
Dehn(G)
is
positive
definite
[cf.
Corollary
5.9,
(iii)].
(iii)
In
general,
the
period
matrix
associated
to
a
profinite
Dehn
multi-twist
may
fail
to
be
nondegenerate
even
if
the
profinite
Dehn
multi-twist
is
nondegenerate.
Indeed,
suppose
that
Σ
=
1,
that
G
is
the
double
[cf.
[CmbGC],
Proposition
2.2,
(i)]
of
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
H
such
that
(Vert(H)
,
Node(H)
,
Cusp(H)
)
=
(1,
0,
2)
.
Suppose,
moreover,
that
H
admits
an
automorphism
which
permutes
the
two
cusps
of
H
and
extends
to
an
automorphism
φ
of
G.
[One
verifies
easily
that
such
data
exist.]
Then
one
may
verify
easily
that
Node(G)
=
2,
that
Cusp(G)
=
0,
and
that
Σ
-module
M
comb
,
hence
also
M
edge
⊗
Σ
M
edge
⊗
Σ
the
free
Z
G
G
G
Z
Z
Σ
)
[cf.
(i)],
is
of
rank
1.
Now
let
us
recall
that
the
Hom
Z
Σ
(Λ
G
,
Z
period
matrix
associated
to
a
positive
definite
profinite
Dehn
multi-twist
is
necessarily
nondegenerate
[cf.
Corollary
5.9,
(iii);
the
proof
of
[CmbGC],
Proposition
2.6].
Thus,
since
Σ
=
1,
it
follows
immediately
from
the
functoriality
of
Ω
G
[cf.
(i)]
and
116
Yuichiro
Hoshi
and
Shinichi
Mochizuki
D
G
[cf.
Theorem
4.8,
(iv)]
with
respect
to
φ
that
the
kernel
of
the
composite
of
natural
homomorphisms
D
G
Ω
G
∼
Σ
)
Λ
G
←−
Dehn(G)
−→
M
G
edge
⊗
Z
Σ
M
G
edge
⊗
Z
Σ
Hom
Z
Σ
(Λ
G
,
Z
Node(G)
Σ
-submodule
of
is
a
free
Z
Node(G)
Λ
G
of
rank
1
that
is
stabi-
lized
by
φ.
On
the
other
hand,
since
profinite
Dehn
multi-twists
of
the
form
(u,
u)
∈
Node(G)
Λ
G
,
where
u
∈
Λ
G
,
are
[mani-
festly!]
positive
definite,
we
thus
conclude
that
the
kernel
in
question
is
equal
to
Λ
G
|
u
∈
Λ
G
}
.
{
(u,
−u)
∈
Node(G)
In
particular,
any
nonzero
element
of
this
kernel
yields
an
ex-
ample
of
a
nondegenerate
profinite
Dehn
multi-twist
whose
as-
sociated
period
matrix
fails
to
be
nondegenerate.
Corollary
5.10
(Combinatorial/group-theoretic
nature
of
scheme-theoreticity
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
Σ
a
nonempty
set
of
prime
numbers;
R
a
complete
discrete
valuation
ring
whose
residue
field
k
is
separably
closed
of
charac-
def
teristic
∈
Σ;
S
log
the
log
scheme
obtained
by
equipping
S
=
Spec
R
with
the
log
structure
determined
by
the
maximal
ideal
of
R;
x
∈
(M
g,r
)
S
(k)
a
k-valued
point
of
the
moduli
stack
of
curves
(M
g,r
)
S
of
type
(g,
r)
the
completion
of
over
S
[cf.
the
discussion
entitled
“Curves”
in
§0).
;
O
log
the
local
ring
of
(M
g,r
)
S
at
the
image
of
x;
T
the
log
scheme
obtained
def
with
the
log
structure
induced
by
the
log
struc-
by
equipping
T
=
Spec
O
log
ture
of
(M
g,r
)
S
[cf.
the
discussion
entitled
“Curves”
in
§0];
t
log
the
log
scheme
obtained
by
equipping
the
closed
point
of
T
with
the
log
struc-
ture
induced
by
the
log
structure
of
T
log
;
X
t
log
the
stable
log
curve
over
log
t
log
corresponding
to
the
natural
strict
(1-)morphism
t
log
→
(M
g,r
)
S
;
I
T
log
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
π
1
(T
log
)
of
T
log
;
I
S
log
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
π
1
(S
log
)
of
S
log
;
G
X
log
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
determined
by
the
stable
log
curve
X
t
log
[cf.
[CmbGC],
Example
2.5];
:
I
T
log
→
Aut(G
X
log
)
the
natural
outer
representation
associated
to
ρ
univ
X
log
t
X
t
log
[cf.
Definition
5.5];
I
a
profinite
group;
ρ
:
I
→
Aut(G
X
log
)
an
Combinatorial
anabelian
topics
I
117
outer
representation
of
pro-Σ
PSC-type
[cf.
[NodNon],
Definition
2.1,
(i)].
Then
the
following
conditions
are
equivalent:
(i)
ρ
is
of
IPSC-type.
(ii)
There
exist
a
morphism
of
log
schemes
φ
log
:
S
log
→
T
log
over
S
and
an
isomorphism
of
outer
representations
of
pro-Σ
∼
◦
I
φ
log
[cf.
[NodNon],
Definition
2.1,
(i)]
PSC-type
ρ
→
ρ
univ
X
t
log
—
where
we
write
I
φ
log
:
I
S
log
→
I
T
log
for
the
homomorphism
induced
by
φ
log
—
i.e.,
there
exist
an
automorphism
β
of
∼
G
X
log
and
an
isomorphism
α
:
I
→
I
S
log
such
that
the
diagram
ρ
−−−−→
I
⏐
⏐
α
ρ
X
Aut(G
X
log
)
⏐
⏐
log
◦I
φ
log
t
I
S
log
−−−
−−−−→
Aut(G
X
log
)
—
where
the
right-hand
vertical
arrow
is
the
automorphism
of
Aut(G
X
log
)
induced
by
β
—
commutes.
(iii)
There
exist
a
morphism
of
log
schemes
φ
log
:
S
log
→
T
log
over
∼
S
and
an
isomorphism
α
:
I
→
I
S
log
such
that
ρ
=
ρ
univ
◦I
φ
log
◦
X
t
log
α
—
where
we
write
I
φ
log
:
I
S
log
→
I
T
log
for
the
homomorphism
induced
by
φ
log
—
i.e.,
the
automorphism
“β”
of
(ii)
may
be
taken
to
be
the
identity.
Proof.
The
equivalence
(i)
⇔
(ii)
follows
from
the
definition
of
the
term
“IPSC-type”
[cf.
[NodNon],
Definition
2.4,
(i)].
The
implication
(iii)
⇒
(ii)
is
immediate.
The
implication
(ii)
⇒
(iii)
follows
immedi-
ately,
in
light
of
the
functoriality
asserted
in
Theorem
4.8,
(iv),
from
Lemma
5.4,
(i),
(ii),
and
Theorem
5.7.
Q.E.D.
Remark
5.10.1.
(i)
The
equivalence
of
Corollary
5.10
essentially
amounts
to
the
equivalence
“IPSC-type
⇐⇒
positive
definite”
which
was
discussed
in
[HM],
Remark
2.14.1,
without
proof.
118
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(ii)
One
way
to
understand
the
equivalence
of
Corollary
5.10
is
as
the
statement
that
the
property
that
an
outer
representation
of
PSC-type
be
of
scheme-theoretic
origin
may
be
formulated
purely
in
terms
of
combinatorics/group
theory.
In
the
final
portion
of
the
present
§5,
we
apply
the
theory
devel-
oped
so
far
[i.e.,
in
particular,
the
equivalences
of
Corollary
5.9,
(ii),
(iii)]
to
derive
results
[cf.
Theorem
5.14]
concerning
normalizers
and
commensurators
of
groups
of
profinite
Dehn
multi-twists.
Definition
5.11.
Let
M
⊆
H
⊆
Out(Π
G
)
be
closed
subgroups
of
Out(Π
G
).
Suppose
further
that
M
is
an
abelian
pro-Σ
group
[such
as
Dehn(G)
—
cf.
Theorem
4.8,
(iv)].
(i)
We
shall
write
scal
(M
)
⊆
N
H
(M
)
⊆
H
N
H
for
the
[closed]
subgroup
of
H
consisting
of
α
∈
H
satisfying
the
following
condition:
α
∈
N
H
(M
),
and,
moreover,
the
action
of
α
on
M
by
conjugation
coincides
with
the
automorphism
of
M
Σ
)
∗
.
We
shall
refer
given
by
multiplication
by
an
element
of
(
Z
scal
(M
)
as
the
scalar-normalizer
of
M
in
H.
to
N
H
(ii)
We
shall
write
scal
(M
)
⊆
C
H
(M
)
⊆
H
C
H
for
the
subgroup
of
H
consisting
of
α
∈
H
satisfying
the
fol-
Σ
-submodule
M
⊆
M
lowing
condition:
there
exists
an
open
Z
α
of
M
[possibly
depending
on
α]
such
that
the
action
of
α
on
H
by
conjugation
determines
an
automorphism
of
M
α
given
Σ
)
∗
.
We
shall
refer
to
by
multiplication
by
an
element
of
(
Z
scal
C
H
(M
)
as
the
scalar-commensurator
of
M
in
H.
Lemma
5.12
(Scalar-normalizers
and
scalar-commensura-
tors).
Let
M
⊆
H
⊆
Out(Π
G
)
be
closed
subgroups
of
Out(Π
G
).
Suppose
further
that
M
is
an
abelian
pro-Σ
group.
Then:
Combinatorial
anabelian
topics
I
(i)
It
holds
that
M
(ii)
119
⊆
Z
H
(M
)
⊆
scal
N
H
(M
)
⊆
scal
C
H
(M
)
.
Σ
-submodule
of
M
,
then
If
M
⊆
M
is
a
Z
scal
scal
scal
scal
(M
)
⊆
N
H
(M
)
;
C
H
(M
)
⊆
C
H
(M
)
.
N
H
If,
moreover,
M
⊆
M
is
open
in
M
,
then
scal
scal
C
H
(M
)
=
C
H
(M
)
.
Proof.
These
assertions
follow
immediately
from
the
various
defi-
nitions
involved.
Q.E.D.
Definition
5.13.
Let
H
⊆
Out(Π
G
)
be
a
closed
subgroup
of
Out(Π
G
).
Then
we
shall
say
that
H
is
IPSC-ample
(respectively,
NN-
ample)
if
H
contains
a
positive
definite
(respectively,
nondegenerate)
[cf.
Definition
5.8]
profinite
Dehn
multi-twist
∈
Dehn(G).
Remark
5.13.1.
It
follows
immediately
from
Theorem
4.8,
(iv),
that
any
open
subgroup
of
Dehn(G)
is
IPSC-ample,
hence
also
NN-ample
[cf.
Definition
5.13].
Theorem
5.14
(Normalizers
and
commensurators
of
groups
of
profinite
Dehn
multi-twists).
Let
Σ
be
a
nonempty
set
of
prime
numbers,
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
Out
C
(Π
G
)
the
group
of
group-theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)]
outomorphisms
of
Π
G
,
and
M
⊆
Out
C
(Π
G
)
a
closed
subgroup
of
Out
C
(Π
G
)
which
is
abelian
pro-Σ.
Then
the
following
hold:
(i)
Suppose
that
one
of
the
following
two
conditions
is
satisfied:
(1)
M
is
IPSC-ample
[cf.
Definition
5.13].
(2)
M
is
NN-ample
[cf.
Definition
5.13],
and
Cusp(G)
=
∅.
120
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Then
it
holds
that
scal
scal
N
Out
C
(Π
)
(M
)
⊆
C
Out
C
(Π
)
(M
)
⊆
Aut(G)
G
G
[cf.
Definition
5.11].
If,
moreover,
M
⊆
Dehn(G)
[cf.
Defini-
tion
4.4],
then
scal
scal
Aut
|Node(G)|
(G)
⊆
N
Out
C
(Π
)
(M
)
⊆
C
Out
C
(Π
)
(M
)
⊆
Aut(G)
G
G
[cf.
Definition
2.6,
(i)].
In
particular,
scal
scal
N
Out
C
(Π
)
(M
)
,
C
Out
C
(Π
)
(M
)
⊆
Aut(G)
G
G
are
open
subgroups
of
Aut(G).
(ii)
If
M
is
an
open
subgroup
of
Dehn(G),
then
it
holds
that
Aut(G)
=
C
Out
C
(Π
G
)
(M
)
.
If,
moreover,
Node(G)
=
∅,
then
Aut
|Node(G)|
(G)
∩
Ker(χ
G
)
=
Z
Out
C
(Π
G
)
(M
)
[cf.
Definition
3.8,
(ii)].
(iii)
It
holds
that
Aut(G)
=
N
Out
C
(Π
G
)
(Dehn(G))
=
C
Out
C
(Π
G
)
(Dehn(G))
.
scal
Proof.
First,
we
verify
the
inclusion
C
Out
C
(Π
)
(M
)
⊆
Aut(G)
as-
G
serted
in
assertion
(i).
Suppose
that
condition
(1)
(respectively,
(2))
is
scal
scal
satisfied.
Let
α
∈
C
Out
C
(Π
)
(M
).
Then
since
α
∈
C
Out
C
(Π
)
(M
),
and
G
G
M
is
IPSC-ample
(respectively,
NN-ample),
it
follows
immediately
that
there
exists
an
element
β
∈
M
of
M
such
that
both
β
and
αβα
−1
=
β
λ
,
Σ
)
∗
,
are
positive
definite
(respectively,
nondegenerate)
profi-
where
λ
∈
(
Z
nite
Dehn
multi-twists.
Thus,
the
graphicity
of
α
follows
immediately
from
[NodNon],
Remark
4.2.1,
together
with
Corollary
5.9,
(iii)
(respec-
tively,
from
[NodNon],
Theorem
A,
together
with
Corollary
5.9,
(ii)).
scal
This
completes
the
proof
of
the
inclusion
C
Out
C
(Π
)
(M
)
⊆
Aut(G),
hence
G
also,
by
Lemma
5.12,
(i),
of
the
two
inclusions
in
the
first
display
of
as-
sertion
(i).
If,
moreover,
M
⊆
Dehn(G),
then
the
inclusion
Aut
|Node(G)|
(G)
⊆
scal
N
Out
C
(Π
)
(M
)
follows
immediately
from
Theorem
4.8,
(v).
Thus,
since
G
Combinatorial
anabelian
topics
I
121
Aut
|Node(G)|
(G)
is
an
open
subgroup
of
Aut(G)
[cf.
Proposition
2.7,
(iii)],
scal
scal
it
follows
immediately
that
N
Out
C
(Π
)
(M
),
hence
also
C
Out
C
(Π
)
(M
),
is
G
G
an
open
subgroup
of
Aut(G).
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
the
equality
Aut(G)
=
C
Out
C
(Π
G
)
(M
)
in
the
first
display
of
assertion
(ii).
It
follows
immediately
from
Theorem
4.8,
(i),
that
Aut(G)
⊆
N
Out
C
(Π
G
)
(Dehn(G))
⊆
C
Out
C
(Π
G
)
(M
).
Thus,
to
verify
the
equality
Aut(G)
=
C
Out
C
(Π
G
)
(M
),
it
suffices
to
verify
the
inclusion
C
Out
C
(Π
G
)
(M
)
⊆
Aut(G).
To
this
end,
let
α
∈
C
Out
C
(Π
G
)
(M
).
Then
it
follows
from
Lemma
5.12,
(ii),
that
scal
scal
−1
scal
−1
)
=
α
·
C
Out
,
C
Out
C
(Π
)
(M
)
=
C
Out
C
(Π
)
(α
·
M
·
α
C
(Π
)
(M
)
·
α
G
G
G
scal
scal
i.e.,
α
∈
N
Out
C
(Π
G
)
(C
Out
Thus,
since
C
Out
C
(Π
)
(M
)).
C
(Π
)
(M
)
is
an
G
G
open
subgroup
of
Aut(G)
[cf.
assertion
(i);
Remark
5.13.1],
we
conclude
that
α
∈
C
Out
C
(Π
G
)
(Aut(G)).
Thus,
the
fact
that
α
∈
Aut(G)
follows
from
the
commensurable
terminality
of
Aut(G)
in
Out(Π
G
),
i.e.,
the
equality
Aut(G)
=
C
Out(Π
G
)
(Aut(G))
[cf.
[CmbGC],
Corollary
2.7,
(iv)].
This
completes
the
proof
of
the
equality
Aut(G)
=
C
Out
C
(Π
G
)
(M
).
Next,
we
verify
the
equality
Aut
|Node(G)|
(G)
∩
Ker(χ
G
)
=
Z
Out
C
(Π
G
)
(M
)
in
the
second
display
of
assertion
(ii).
Now
it
follows
immediately
from
Theorem
4.8,
(v),
that
Aut
|Node(G)|
(G)
∩
Ker(χ
G
)
⊆
Z
Out
C
(Π
G
)
(M
).
Thus,
to
show
the
equality
in
question,
it
suffices
to
verify
the
inclu-
sion
Z
Out
C
(Π
G
)
(M
)
⊆
Aut
|Node(G)|
(G)
∩
Ker(χ
G
).
To
this
end,
let
us
observe
that
since
Aut(G)
=
C
Out
C
(Π
G
)
(M
)
[cf.
the
preceding
para-
graph],
it
holds
that
Z
Out
C
(Π
G
)
(M
)
⊆
Aut(G).
Thus,
since
the
action
of
Z
Out
C
(Π
G
)
(M
)
on
M
by
conjugation
preserves
and
induces
the
iden-
tity
automorphism
on
the
intersection
of
M
with
each
direct
summand
D
G
∼
of
e∈Node(G)
Λ
G
←
Dehn(G)
[i.e.,
each
“Λ
G
”],
it
follows
immediately
from
Theorem
4.8,
(v),
in
light
of
our
assumption
that
Node(G)
=
∅,
that
Z
Out
C
(Π
G
)
(M
)
⊆
Aut
|Node(G)|
(G)
∩
Ker(χ
G
).
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
assertion
(ii),
together
with
Theorem
4.8,
(i).
This
completes
the
proof
of
Theorem
5.14.
Q.E.D.
Remark
5.14.1.
In
the
notation
of
Theorem
5.14,
(i)
(respectively,
Theorem
5.14,
(ii)),
in
general,
the
inclusion
scal
C
Out
C
(Π
)
(M
)
⊆
Aut(G)
G
122
Yuichiro
Hoshi
and
Shinichi
Mochizuki
[hence,
a
fortiori,
by
the
inclusions
of
the
first
display
of
Theorem
5.14,
scal
(i),
the
inclusion
N
Out
C
(Π
)
(M
)
⊆
Aut(G)]
(respectively,
in
general,
the
G
inclusion
N
Out
C
(Π
G
)
(M
)
⊆
Aut(G)
)
is
strict.
Indeed,
suppose
that
there
exist
a
node
e
∈
Node(G)
and
an
automorphism
α
∈
Aut(G)
of
G
such
that
α
does
not
stabilize
e,
and
χ
G
(α)
=
1.
[For
example,
in
the
notation
of
the
final
paragraph
of
the
proof
of
Theorem
5.7,
the
node
e
1
and
the
automorphism
induced
by
τ
of
G
X
log
satisfy
these
conditions.]
Now
fix
a
prime
number
l
∈
Σ;
write
def
M
=
l
·
(Λ
G
)
e
⊕
f
=e
Λ
G
⊆
D
G
∼
Λ
G
←
Dehn(G)
f
∈Node(G)
—
where
we
use
the
notation
(Λ
G
)
e
to
denote
a
copy
of
Λ
G
indexed
by
e
∈
Node(G).
Then
M
is
an
open
subgroup
of
Dehn(G),
hence
also
IPSC-
ample
[cf.
Remark
5.13.1],
but
it
follows
immediately
from
Theorem
4.8,
scal
(v),
that
α
∈
C
Out
C
(Π
)
(M
)
(respectively,
α
∈
N
Out
C
(Π
G
)
(M
)).
G
§6.
Centralizers
of
geometric
monodromy
In
the
present
§,
we
study
the
centralizer
of
the
image
of
certain
geometric
monodromy
groups.
As
an
application,
we
prove
a
“geometric
version
of
the
Grothendieck
conjecture”
for
the
universal
curve
over
the
moduli
stack
of
pointed
smooth
curves
[cf.
Theorem
6.13
below].
Definition
6.1.
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
Π
a
pro-Σ
surface
group
[cf.
[MT],
Definition
1.2].
Then
we
shall
write
Out
C
(Π)
=
Out
FC
(Π)
=
Out
PFC
(Π)
for
the
group
of
outomorphisms
of
Π
which
induce
bijections
on
the
set
of
cuspidal
inertia
subgroups
of
Π.
We
shall
refer
to
an
element
of
Out
C
(Π)
=
Out
FC
(Π)
=
Out
PFC
(Π)
as
a
C-,
FC-,
or
PFC-admissible
outomorphism
of
Π.
Remark
6.1.1.
In
the
notation
of
Definition
6.1,
suppose
that
ei-
ther
Σ
=
1
or
Σ
=
Primes.
Then
it
follows
from
the
various
def-
initions
involved
that
Π
is
equipped
with
a
natural
structure
of
pro-
Σ
configuration
space
group
[cf.
[MT],
Definition
2.3,
(i)].
Thus,
the
Combinatorial
anabelian
topics
I
123
terms
“C-/FC-/PFC-admissible
outomorphism
of
Π”
and
the
notation
“Out
C
(Π)
=
Out
FC
(Π)”
have
already
been
defined
in
[CmbCsp],
Defini-
tion
1.1,
(ii),
and
Definition
1.4,
(iii),
of
the
present
paper.
In
this
case,
however,
one
may
verify
easily
that
these
definitions
coincide.
Lemma
6.2
(Extensions
arising
from
log
configuration
spaces).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g−2+r
>
0;
0
<
m
<
n
positive
integers;
Σ
F
⊆
Σ
B
nonempty
sets
of
prime
numbers;
k
an
algebraically
closed
field
of
characteristic
zero;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
given
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
Suppose
that
Σ
F
⊆
Σ
B
satisfy
one
of
the
following
two
conditions:
(1)
Σ
F
and
Σ
B
determine
PT-formations
[i.e.,
are
either
of
car-
dinality
one
or
equal
to
Primes
—
cf.
[MT],
Definition
1.1,
(ii)].
(2)
n
−
m
=
1
and
Σ
B
=
Primes.
Write
log
X
n
log
,
X
m
for
the
n-th,
m-th
log
configuration
spaces
of
the
stable
log
curve
X
log
def
[cf.
the
discussion
entitled
“Curves”
in
§0],
respectively;
Π
n
,
Π
B
=
Π
m
for
the
respective
maximal
pro-Σ
B
quotients
of
the
kernels
of
the
natu-
log
)
π
1
((Spec
k)
log
);
ral
surjections
π
1
(X
n
log
)
π
1
((Spec
k)
log
),
π
1
(X
m
Π
n/m
⊆
Π
n
for
the
kernel
of
the
surjection
Π
n
Π
B
=
Π
m
induced
log
obtained
by
forgetting
the
last
(n
−
m)
by
the
projection
X
n
log
→
X
m
factors;
Π
F
for
the
maximal
pro-Σ
F
quotient
of
Π
n/m
;
Π
T
for
the
quo-
tient
of
Π
n
by
the
kernel
of
the
natural
surjection
Π
n/m
Π
F
.
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
F
−→
Π
T
−→
Π
B
−→
1
,
which
determines
an
outer
representation
ρ
n/m
:
Π
B
−→
Out(Π
F
)
.
Then
the
following
hold:
(i)
The
isomorphism
class
of
the
exact
sequence
of
profinite
groups
1
−→
Π
F
−→
Π
T
−→
Π
B
−→
1
124
Yuichiro
Hoshi
and
Shinichi
Mochizuki
depends
only
on
(g,
r)
and
the
pair
(Σ
F
,
Σ
B
),
i.e.,
if
1
→
Π
•
F
→
Π
•
T
→
Π
•
B
→
1
is
the
exact
sequence
“1
→
Π
F
→
Π
T
→
Π
B
→
1”
associated,
with
respect
to
the
same
(Σ
F
,
Σ
B
),
to
another
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
,
then
there
exists
a
commutative
diagram
of
profinite
groups
1
−−−−→
Π
F
−−−−→
Π
T
−−−−→
Π
B
−−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
1
−−−−→
Π
•
F
−−−−→
Π
•
T
−−−−→
Π
•
B
−−−−→
1
—
where
the
vertical
arrows
are
isomorphisms
which
may
be
chosen
to
arise
scheme-theoretically.
(ii)
The
profinite
group
Π
B
is
equipped
with
a
natural
structure
of
pro-Σ
B
configuration
space
group
[cf.
[MT],
Definition
2.3,
(i)].
If,
moreover,
Σ
F
⊆
Σ
B
satisfies
condition
(1)
(re-
spectively,
(2)),
then
the
profinite
group
Π
F
is
equipped
with
a
natural
structure
of
pro-Σ
F
configuration
space
group
(re-
spectively,
surface
group
[cf.
[MT],
Definition
1.2]).
(iii)
The
outer
representation
ρ
n/m
:
Π
B
→
Out(Π
F
)
factors
through
the
closed
subgroup
Out
C
(Π
F
)
⊆
Out(Π
F
)
[cf.
Definition
6.1;
[CmbCsp],
Definition
1.1,
(ii)].
Proof.
Assertion
(i)
follows
immediately
by
considering
a
suitable
specialization
isomorphism
[cf.
the
discussion
preceding
[CmbCsp],
Def-
inition
2.1,
as
well
as
Remark
5.6.1
of
the
present
paper].
Assertion
(ii)
follows
immediately
from
assertion
(i),
together
with
the
various
def-
initions
involved.
Assertion
(iii)
follows
immediately
from
the
various
definitions
involved.
This
completes
the
proof
of
Lemma
6.2.
Q.E.D.
Definition
6.3.
In
the
notation
of
Lemma
6.2
in
the
case
where
(m,
n,
Σ
B
)
=
(1,
2,
Primes),
let
x
∈
X(k)
be
a
k-valued
point
of
the
underlying
scheme
X
of
X
log
.
(i)
We
shall
denote
by
G
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
deter-
mined
by
the
stable
log
curve
X
log
;
by
G
x
Combinatorial
anabelian
topics
I
125
the
semi-graph
of
anabelioids
of
pro-Σ
F
PSC-type
determined
def
by
the
geometric
fiber
of
X
2
log
→
X
log
over
x
log
=
x
×
X
X
log
;
by
Π
G
,
Π
G
x
the
[pro-Primes,
pro-Σ
F
]
fundamental
groups
of
G,
G
x
,
respectively.
Thus,
we
have
a
natural
outer
isomorphism
∼
Π
B
−→
Π
G
and
a
natural
Im(ρ
2/1
)
(⊆
Out(Π
F
))-torsor
of
outer
isomor-
phisms
∼
Π
F
−→
Π
G
x
.
∼
∼
Let
us
fix
isomorphisms
Π
B
→
Π
G
,
Π
F
→
Π
G
x
that
belong
to
these
collections
of
isomorphisms.
(ii)
Denote
by
c
F
diag,x
∈
Cusp(G
x
)
the
cusp
of
G
x
[i.e.,
the
cusp
of
the
geometric
fiber
of
X
2
log
→
X
log
over
x
log
]
determined
by
the
diagonal
divisor
of
X
2
log
.
For
v
∈
Vert(G)
(respectively,
c
∈
Cusp(G))
[i.e.,
which
cor-
responds
to
an
irreducible
component
(respectively,
a
cusp)
of
X
log
],
denote
by
v
x
F
∈
Vert(G
x
)
(respectively,
c
F
x
∈
Cusp(G
x
))
the
vertex
(respectively,
cusp)
of
G
x
that
corresponds
naturally
to
v
∈
Vert(G)
(respectively,
c
∈
Cusp(G)).
(iii)
Let
e
∈
Edge(G),
v
∈
Vert(G),
S
⊆
VCN(G),
and
z
∈
VCN(G).
Then
we
shall
say
that
x
lies
on
e
if
the
image
of
x
is
the
cusp
or
node
corresponding
to
e
∈
Edge(G).
We
shall
say
that
x
lies
on
v
if
x
does
not
lie
on
any
edge
of
G,
and,
moreover,
the
image
of
x
is
contained
in
the
irreducible
component
corresponding
to
v
∈
Vert(G).
We
shall
write
x
S
if
x
lies
on
some
s
∈
S.
We
shall
write
x
z
if
x
{z}.
Lemma
6.4
(Cusps
and
vertices
of
fibers).
In
the
notation
of
Definition
6.3,
let
x,
x
∈
X(k)
be
k-valued
points
of
X.
Then
the
following
hold:
(i)
∼
The
isomorphism
Π
G
x
→
Π
G
x
obtained
by
forming
the
compos-
∼
∼
ite
of
the
isomorphisms
Π
G
x
←
Π
F
→
Π
G
x
[cf.
Definition
6.3,
126
Yuichiro
Hoshi
and
Shinichi
Mochizuki
(i)]
is
group-theoretically
cuspidal
[cf.
[CmbGC],
Defini-
tion
1.4,
(iv)].
(ii)
The
injection
Cusp(G)
→
Cusp(G
x
)
given
by
mapping
c
→
c
F
x
determines
a
bijection
∼
Cusp(G)
−→
Cusp(G
x
)
\
{c
F
diag,x
}
[cf.
Definition
6.3,
(ii)].
Moreover,
if
we
regard
Cusp(G)
as
a
subset
of
each
of
the
sets
Cusp(G
x
),
Cusp(G
x
)
by
means
of
the
∼
above
injections,
then
the
bijection
Cusp(G
x
)
→
Cusp(G
x
)
de-
termined
by
the
group-theoretically
cuspidal
isomorphism
∼
F
Π
G
x
→
Π
G
x
of
(i)
maps
c
F
diag,x
→
c
diag,x
and
induces
the
identity
automorphism
on
Cusp(G).
Thus,
in
the
remain-
der
of
the
present
§,
we
shall
omit
the
subscript
“x”
from
the
F
notation
“c
F
x
”
and
“c
diag,x
”.
(iii)
The
injection
Vert(G)
→
Vert(G
x
)
given
by
mapping
v
→
v
x
F
[cf.
Definition
6.3,
(ii)]
is
bijective
if
and
only
if
x
Vert(G)
[cf.
Definition
6.3,
(iii)].
If
x
Edge(G),
then
the
comple-
ment
of
the
image
of
Vert(G)
in
Vert(G
x
)
is
of
cardinality
one;
in
this
case,
we
shall
write
F
∈
Vert(G
x
)
\
Vert(G)
v
new,x
for
the
unique
element
of
Vert(G
x
)
\
Vert(G).
(iv)
Suppose
that
x
Cusp(G)
(respectively,
Node(G)).
Then
F
F
it
holds
that
c
F
diag
∈
C(v
new,x
)
[cf.
(iii)],
and
(C(v
new,x
)
,
F
)
)
=
(2,
1)
(respectively,
=
(1,
2)).
Moreover,
for
any
N
(v
new,x
F
element
e
F
∈
N
(v
new,x
),
it
holds
that
V(e
F
)
=
2.
Proof.
These
assertions
follow
immediately
from
the
various
defi-
nitions
involved.
Q.E.D.
Definition
6.5.
In
the
notation
of
Definition
6.3:
(i)
Write
def
Cusp
F
(G)
=
Cusp(G)
{c
F
diag
}
[cf.
Definition
6.3,
(ii);
Lemma
6.4,
(ii)].
Combinatorial
anabelian
topics
I
127
(ii)
Let
α
∈
Out
C
(Π
F
)
be
an
C-admissible
outomorphism
of
Π
F
[cf.
Definition
6.1;
Lemma
6.2,
(ii)].
Then
it
follows
immediately
from
Lemma
6.4,
(ii),
that
for
any
k-valued
point
x
∈
X(k)
of
X,
the
automorphism
of
Cusp
F
(G)
[cf.
(i)]
obtained
by
conjugating
the
natural
action
of
α
on
Cusp(G
x
)
by
the
natural
∼
bijection
Cusp
F
(G)
→
Cusp(G
x
)
implicit
in
Lemma
6.4,
(ii),
does
not
depend
on
the
choice
of
x.
We
shall
refer
to
this
automorphism
of
Cusp
F
(G)
as
the
automorphism
of
Cusp
F
(G)
determined
by
α.
Thus,
we
have
a
natural
homomorphism
Out
C
(Π
F
)
→
Aut(Cusp
F
(G)).
(iii)
For
c
∈
Cusp
F
(G)
[cf.
(i)],
we
shall
refer
to
a
closed
subgroup
∼
of
Π
F
obtained
as
the
image
—
via
the
isomorphism
Π
G
x
←
Π
F
[cf.
Definition
6.3,
(i)]
for
some
k-valued
point
x
∈
X(k)
—
of
a
cuspidal
subgroup
of
Π
G
x
associated
to
the
cusp
of
G
x
corresponding
to
c
∈
Cusp
F
(G)
as
a
cuspidal
subgroup
of
Π
F
associated
to
c
∈
Cusp
F
(G).
Note
that
it
follows
immediately
from
Lemma
6.4,
(ii),
that
the
Π
F
-conjugacy
class
of
a
cuspidal
subgroup
of
Π
F
associated
to
c
∈
Cusp
F
(G)
depends
only
on
c
∈
Cusp
F
(G),
i.e.,
does
not
depend
on
the
choice
of
x
or
on
the
choices
of
isomorphisms
made
in
Definition
6.3,
(i).
Lemma
6.6
(Images
of
VCN-subgroups
of
fibers).
In
the
no-
tation
of
Definition
6.3,
let
Π
c
Fdiag
⊆
Π
F
be
a
cuspidal
subgroup
of
Π
F
F
associated
to
c
F
diag
∈
Cusp
(G)
[cf.
Definition
6.5,
(i),
(iii)],
x
∈
X(k)
F
a
k-valued
point
of
X,
z
∈
VCN(G
x
)
\
{c
F
diag
},
and
Π
z
F
⊆
Π
G
x
a
VCN-
F
subgroup
of
Π
G
x
associated
to
z
.
Write
N
diag
⊆
Π
F
for
the
normal
closed
subgroup
of
Π
F
topologically
normally
generated
by
Π
c
Fdiag
.
[Note
that
it
follows
immediately
from
Lemma
6.4,
(i),
that
N
diag
is
normal
in
Π
T
.]
Then
the
following
hold:
(i)
Write
G
Σ
F
for
the
semi-graph
of
anabelioids
of
pro-Σ
F
PSC-
type
obtained
by
forming
the
pro-Σ
F
completion
of
G
[cf.
[SemiAn],
Definition
2.9,
(ii)].
Then
there
exists
a
natural
∼
outer
isomorphism
Π
F
/N
diag
→
Π
G
ΣF
that
satisfies
the
fol-
lowing
conditions:
•
Suppose
that
x
Vert(G)
[cf.
Definition
6.3,
(iii)].
Then
the
Π
G
ΣF
-conjugacy
class
of
the
image
of
the
composite
∼
∼
Π
z
F
→
Π
G
x
←
Π
F
Π
F
/N
diag
→
Π
G
ΣF
128
Yuichiro
Hoshi
and
Shinichi
Mochizuki
coincides
with
the
Π
G
ΣF
-conjugacy
class
of
any
VCN-sub-
group
of
Π
G
ΣF
associated
to
the
element
of
VCN(G
Σ
F
)
=
VCN(G)
naturally
determined
by
z
F
.
•
F
Suppose
that
x
e
∈
Edge(G),
and
that
z
F
∈
{v
new,x
}∪
F
F
F
F
E(v
new,x
)
(respectively,
z
∈
{v
new,x
}
∪
E(v
new,x
))
[cf.
Lemma
6.4,
(iii)].
Then
the
Π
G
ΣF
-conjugacy
class
of
the
image
of
the
composite
∼
∼
Π
z
F
→
Π
G
x
←
Π
F
Π
F
/N
diag
→
Π
G
ΣF
coincides
with
the
Π
G
ΣF
-conjugacy
class
of
any
VCN-sub-
group
of
Π
G
ΣF
associated
to
the
element
of
VCN(G
Σ
F
)
=
VCN(G)
natural
determined
by
z
F
(respectively,
associated
to
e
∈
Edge(G
Σ
F
)
=
Edge(G)).
(ii)
The
image
of
the
composite
∼
Π
z
F
→
Π
G
x
←
Π
F
Π
F
/N
diag
is
commensurably
terminal.
(iii)
Suppose
that
either
•
z
F
∈
Edge(G
x
),
or
•
z
F
=
v
x
F
for
v
∈
Vert(G)
such
that
x
does
not
lie
on
v.
Then
the
composite
∼
Π
z
F
→
Π
G
x
←
Π
F
Π
F
/N
diag
is
injective.
(iv)
Let
Π
(z
)
F
⊆
Π
G
x
be
a
VCN-subgroup
of
Π
G
x
associated
to
an
element
(z
)
F
∈
VCN(G
x
)
\
{c
F
diag
}.
Suppose
that
either
•
x
Vert(G),
or
•
F
F
x
Edge(G),
and
z
F
,
(z
)
F
∈
{v
new,x
}
∪
E(v
new,x
).
Then
if
the
Π
F
/N
diag
-conjugacy
classes
of
the
images
of
Π
z
F
,
Π
(z
)
F
⊆
Π
G
x
via
the
composite
∼
Π
G
x
←
Π
F
Π
F
/N
diag
Combinatorial
anabelian
topics
I
129
coincide,
then
z
F
=
(z
)
F
.
Proof.
Assertion
(i)
follows
immediately
from
the
various
defini-
tions
involved.
Assertion
(ii)
follows
immediately
from
[CmbGC],
Propo-
sition
1.2,
(ii),
and
assertion
(i),
together
with
the
various
definitions
involved.
Assertion
(iii)
follows
immediately
from
assertion
(i),
together
with
the
various
definitions
involved.
Assertion
(iv)
follows
immediately
from
[CmbGC],
Proposition
1.2,
(i),
and
assertion
(i),
together
with
the
various
definitions
involved.
Q.E.D.
Lemma
6.7
(Outomorphisms
preserving
the
diagonal).
In
the
notation
of
Definition
6.3,
let
H
⊆
Π
B
be
an
open
subgroup
of
Π
B
,
def
α
an
automorphism
of
Π
T
|
H
=
Π
T
×
Π
B
H
over
H,
α
F
∈
Out(Π
F
)
the
|
Π
F
of
α
to
Π
F
⊆
outomorphism
of
Π
F
determined
by
the
restriction
α
Π
T
|
H
,
and
Π
c
Fdiag
⊆
Π
F
a
cuspidal
subgroup
of
Π
F
associated
to
c
F
diag
∈
Cusp
F
(G)
[cf.
Definition
6.5,
(i),
(iii)].
Then
the
following
hold:
(i)
Suppose
that
α
preserves
Π
c
Fdiag
⊆
Π
F
.
Then
the
automorphism
of
Π
F
/N
diag
[where
we
refer
to
the
statement
of
Lemma
6.6
is
the
identity
automor-
concerning
N
diag
]
induced
by
α
phism.
If,
moreover,
α
F
is
C-admissible
[cf.
Definition
6.1;
Lemma
6.2,
(ii)],
then
the
automorphism
of
Cusp
F
(G)
induced
by
α
F
[cf.
Definition
6.5,
(ii)]
is
the
identity
automor-
phism.
(ii)
Let
e
∈
Edge(G),
x
∈
X(k)
be
such
that
x
e.
Suppose
that
α
F
is
C-admissible,
and
that
Edge(G)
=
{e}
∪
Cusp(G).
∼
Then
it
holds
that
α
F
∈
Aut(G
x
)
(⊆
Out(Π
G
x
)
←
Out(Π
F
)).
If,
moreover,
α
preserves
Π
c
Fdiag
⊆
Π
F
,
then
α
F
∈
Aut
|grph|
(G
x
)
(⊆
Aut(G
x
)).
Proof.
First,
we
verify
assertion
(i).
Now
let
us
observe
that
it
follows
immediately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
[CmbCsp],
Proposition
1.2,
(iii)
—
i.e.,
by
considering
the
action
of
α
on
the
decomposition
subgroup
D
⊆
Π
T
|
H
of
Π
T
|
H
associated
to
the
diagonal
divisor
of
X
2
log
such
that
Π
c
Fdiag
⊆
D,
and
applying
the
induces
the
identity
au-
fact
that
D
=
N
Π
T
|
H
(Π
c
Fdiag
)
⊆
Π
T
|
H
—
that
α
tomorphism
on
some
normal
open
subgroup
J
⊆
Π
F
/N
diag
of
Π
F
/N
diag
.
Thus,
it
follows
immediately
from
the
slimness
[cf.
[CmbGC],
Remark
130
Yuichiro
Hoshi
and
Shinichi
Mochizuki
∼
1.1.3]
of
Π
G
ΣF
←
Π
F
/N
diag
→
Aut(J)
that
α
induces
the
identity
au-
tomorphism
on
Π
F
/N
diag
.
This
completes
the
proof
of
the
fact
that
α
induces
the
identity
automorphism
of
Π
F
/N
diag
.
On
the
other
hand,
if,
induces
the
identity
automor-
moreover,
α
F
is
C-admissible,
then
since
α
phism
of
Π
F
/N
diag
,
it
follows
immediately
from
[CmbGC],
Proposition
∼
1.2,
(i),
applied
to
the
cuspidal
inertia
subgroups
of
Π
F
/N
diag
→
Π
G
ΣF
F
[cf.
Lemma
6.6,
(i)]
that
the
automorphism
of
Cusp
(G)
induced
by
α
F
is
the
identity
automorphism.
This
completes
the
proof
of
assertion
(i).
∼
Next,
we
verify
assertion
(ii).
Let
Π
e
⊆
Π
G
←
Π
B
be
an
edge-
like
subgroup
associated
to
the
edge
e
∈
Edge(G).
By
abuse
of
nota-
tion,
we
shall
write
H
∩
Π
e
⊆
Π
B
for
the
intersection
of
H
with
the
is
an
au-
image
of
Π
e
in
Π
B
.
Now
since
α
F
is
C-admissible,
and
α
tomorphism
of
Π
T
|
H
over
H,
it
holds
that
α
F
∈
Z
Out
C
(Π
F
)
(ρ
2/1
(H))
[cf.
the
discussion
entitled
“Topological
groups”
in
§0],
hence
also
that
α
F
∈
Z
Out
C
(Π
F
)
(ρ
2/1
(H
∩
Π
e
)).
On
the
other
hand,
in
light
of
the
well-known
structure
of
X
log
in
a
neighborhood
of
the
cusp
or
node
cor-
responding
to
e,
one
verifies
easily
—
by
applying
[NodNon],
Proposition
2.14,
together
with
our
assumption
that
Edge(G)
=
{e}
∪
Cusp(G)
—
that
the
image
of
the
composite
∼
ρ
2/1
∼
Π
e
→
Π
G
←
Π
B
→
Out(Π
F
)
→
Out(Π
G
x
)
,
∼
hence
also
the
image
ρ
2/1
(H∩Π
e
)
⊆
Out(Π
F
)
→
Out(Π
G
x
),
is
NN-ample
[cf.
Definition
5.13;
Theorem
5.9,
(ii)].
Thus,
since
c
F
diag
∈
Cusp(G
x
)
=
∅,
it
follows
immediately
from
Theorem
5.14,
(i),
that
α
F
∈
Aut(G
x
).
This
completes
the
proof
of
the
fact
that
α
F
∈
Aut(G
x
).
Now
suppose,
moreover,
that
α
preserves
Π
c
Fdiag
⊆
Π
F
.
Then
it
follows
from
assertion
F
.
On
the
(i)
that
α
F
fixes
the
cusps
of
G
x
,
hence
that
it
fixes
v
new,x
other
hand,
since
α
induces
the
identity
automorphism
of
Π
F
/N
diag
[cf.
assertion
(i)],
it
follows
from
Lemma
6.6,
(iii),
(iv),
that
α
F
fixes
the
F
,
as
well
as
[cf.
[CmbGC],
Proposition
vertices
of
G
x
that
are
=
v
new,x
1.2,
(i)]
the
branches
of
nodes
of
G
x
that
abut
to
such
vertices.
Thus,
α
F
∈
Aut
|grph|
(G
x
),
as
desired.
This
completes
the
proof
of
assertion
(ii).
Q.E.D.
Lemma
6.8
(Triviality
of
certain
outomorphisms).
In
the
no-
tation
of
Definition
6.3,
let
Π
c
Fdiag
⊆
Π
F
be
a
cuspidal
subgroup
of
Π
F
F
associated
to
c
F
diag
∈
Cusp
(G)
[cf.
Definition
6.5,
(i),
(iii)],
H
⊆
Π
B
an
open
subgroup
of
Π
B
,
and
α
∈
Z
Out
C
(Π
F
)
(ρ
2/1
(H))
[cf.
Definition
6.1;
Combinatorial
anabelian
topics
I
131
Lemma
6.2,
(ii)].
Suppose
that
α
preserves
the
Π
F
-conjugacy
class
of
Π
c
Fdiag
⊆
Π
F
.
Then
α
is
the
identity
outomorphism.
Proof.
The
following
argument
is
essentially
the
same
as
the
argu-
ment
applied
in
[CmbCsp],
[NodNon]
to
prove
[CmbCsp],
Corollary
2.3,
(ii);
[NodNon],
Corollary
5.3.
def
∈
Aut
H
(Π
T
|
H
)
a
lifting
of
α
∈
Let
Π
T
|
H
=
Π
T
×
Π
B
H
and
α
∼
Z
Out
C
(Π
F
)
(ρ
2/1
(H))
⊆
Z
Out(Π
F
)
(ρ
2/1
(H))
←
Aut
H
(Π
T
|
H
)/Inn(Π
F
)
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Since
we
have
as-
sumed
that
α
preserves
the
Π
F
-conjugacy
class
of
Π
c
Fdiag
⊆
Π
F
,
it
fol-
lows
from
Lemma
6.7,
(i),
(ii),
that
by
replacing
α
by
a
suitable
Π
F
-
conjugate
of
α
,
we
may
assume
without
loss
of
generality
that
α
pre-
serves
Π
c
Fdiag
⊆
Π
F
,
and,
moreover,
that
(a)
the
automorphism
of
Π
F
/N
diag
induced
by
α
is
the
identity
automorphism;
(b)
for
e
∈
Edge(G),
x
∈
X(k)
such
that
x
e,
if
Edge(G)
=
{e}
∼
∪
Cusp(G),
then
α
∈
Aut
|grph|
(G
x
)
(⊆
Out(Π
G
x
)
←
Out(Π
F
)).
Next,
we
claim
that
(∗
1
):
if
(g,
r)
=
(0,
3),
then
α
is
the
identity
outomor-
phism.
Indeed,
write
c
1
,
c
2
,
c
3
∈
Cusp(G)
for
the
three
distinct
cusps
of
G;
v
∈
Vert(G)
for
the
unique
vertex
of
G.
For
i
∈
{1,
2,
3},
let
x
i
∈
X(k)
be
such
that
x
i
c
i
.
Next,
let
us
observe
that
since
our
assumption
that
(g,
r)
=
(0,
3)
implies
that
Node(G)
=
∅,
it
follows
immediately
∼
from
(b)
that
for
i
∈
{1,
2,
3},
the
outomorphism
α
of
Π
G
xi
←
Π
F
is
∼
∈
Aut
|grph|
(G
x
i
)
(⊆
Out(Π
G
xi
)
←
Out(Π
F
)).
Next,
let
us
fix
a
verticial
∼
subgroup
Π
v
x
F
⊆
Π
G
x
2
←
Π
F
associated
to
v
x
F
2
∈
Vert(G
x
2
)
[cf.
Defini-
2
tion
6.3,
(ii)].
Then
since
α
∈
Aut
|grph|
(G
x
2
),
it
follows
immediately
from
the
commensurable
terminality
of
the
image
of
the
composite
Π
v
x
F
→
2
∼
Π
G
x
2
←
Π
F
Π
F
/N
diag
[cf.
Lemma
6.6,
(ii)],
together
with
(a),
that
v
F
)
=
Π
v
F
.
Thus,
there
exists
an
N
diag
-conjugate
β
of
α
such
that
β(Π
x
x
2
∼
2
since
the
composite
Π
v
x
F
→
Π
G
x
2
←
Π
F
Π
F
/N
diag
is
injective
[cf.
2
Lemma
6.6,
(iii)],
it
follows
immediately
from
(a)
that
β
induces
the
∼
identity
automorphism
on
Π
v
x
F
⊆
Π
G
x
2
←
Π
F
.
Next,
let
Π
c
F1
⊆
Π
F
be
a
2
cuspidal
subgroup
of
Π
F
associated
to
c
1
∈
Cusp
F
(G)
[cf.
Definition
6.5,
∼
∼
(iii)]
which
is
contained
in
Π
v
x
F
⊆
Π
G
x
2
←
Π
F
;
Π
v
x
F
⊆
Π
G
x
3
←
Π
F
a
2
3
132
Yuichiro
Hoshi
and
Shinichi
Mochizuki
verticial
subgroup
associated
to
v
x
F
3
∈
Vert(G
x
3
)
that
contains
Π
c
F1
⊆
Π
F
.
∼
Then
since
β
induces
the
identity
automorphism
on
Π
v
F
⊆
Π
G
←
Π
F
,
x
2
x
2
c
F
)
=
Π
c
F
.
Thus,
since
it
follows
from
the
inclusion
Π
c
F1
⊆
Π
v
x
F
that
β(Π
1
1
2
∼
the
verticial
subgroup
Π
v
x
F
⊆
Π
G
x
3
←
Π
F
is
the
unique
verticial
sub-
3
∼
group
of
Π
G
x
3
←
Π
F
associated
to
v
x
F
3
∈
Vert(G
x
3
)
which
contains
Π
c
F1
[cf.
[CmbGC],
Proposition
1.5,
(i)],
it
follows
immediately
from
the
fact
v
F
)
=
Π
v
F
.
In
particular,
since
the
that
α
∈
Aut
|grph|
(G
x
3
)
that
β(Π
x
3
x
3
composite
Π
v
x
F
→
Π
F
Π
F
/N
diag
is
injective
[cf.
Lemma
6.6,
(iii)],
it
3
follows
immediately
from
(a)
that
β
induces
the
identity
automorphism
∼
on
Π
v
x
F
⊆
Π
G
x
3
←
Π
F
.
On
the
other
hand,
since
Π
F
is
topologically
3
∼
∼
generated
by
Π
v
x
F
⊆
Π
G
x
2
←
Π
F
and
Π
v
x
F
⊆
Π
G
x
3
←
Π
F
[cf.
[CmbCsp],
2
3
Lemma
1.13],
this
implies
that
β
induces
the
identity
automorphism
on
Π
F
.
This
completes
the
proof
of
the
claim
(∗
1
).
Next,
we
claim
that
(∗
2
):
for
arbitrary
(g,
r),
α
is
the
identity
outomor-
phism.
Indeed,
we
verify
the
claim
(∗
2
)
by
induction
on
3g
−3+r.
If
3g
−3+r
=
0,
i.e.,
(g,
r)
=
(0,
3),
then
the
claim
(∗
2
)
amounts
to
the
claim
(∗
1
).
Now
suppose
that
3g
−3+r
>
1,
and
that
the
induction
hypothesis
is
in
force.
Since
3g
−
3
+
r
>
1,
one
verifies
easily
that
there
exists
a
stable
log
curve
Y
log
of
type
(g,
r)
over
(Spec
k)
log
such
that
Y
log
has
precisely
one
node.
Thus,
it
follows
immediately
from
Lemma
6.2,
(i),
that
to
verify
the
claim
(∗
2
),
by
replacing
X
log
by
Y
log
,
we
may
assume
without
loss
of
generality
that
Node(G)
=
1.
Let
e
be
the
unique
node
of
G
and
x
∈
X(k)
such
that
x
e.
Now
let
us
observe
that
since
Node(G)
=
1,
and
e
∈
Node(G),
it
follows
from
(b)
that
α
∈
Aut
|grph|
(G
x
)
(⊆
∼
F
F
Out(Π
G
x
)
←
Out(Π
F
)).
Write
{e
F
1
,
e
2
}
=
N
(v
new,x
)
[cf.
Lemma
6.4,
(iv)].
Also,
for
i
∈
{1,
2},
denote
by
v
i
∈
Vert(G)
the
vertex
of
G
F
F
such
that
(v
i
)
F
x
∈
Vert(G
x
)
is
the
unique
element
of
V(e
i
)
\
{v
new,x
}
[cf.
Lemma
6.4,
(iv)];
by
H
i
the
sub-semi-graph
of
PSC-type
of
the
F
,
(v
i
)
F
underlying
semi-graph
G
x
of
G
x
whose
set
of
vertices
=
{v
new,x
x
};
def
and
by
S
i
=
Node((G
x
)|
H
i
)
\
{e
F
i
}
⊆
Node((G
x
)|
H
i
)
the
complement
}.
[Thus,
if
G
is
noncyclically
primitive
(respectively,
cyclically
of
{e
F
i
primitive)
[cf.
Definition
4.1],
then
H
i
=
G
x
and
S
i
=
∅
(respectively,
H
i
=
G
x
and
S
i
=
{e
F
3−i
}).
In
particular,
S
i
⊆
Node((G
x
)|
H
i
)
is
not
of
separating
type.]
Next,
let
us
observe
that
to
complete
the
proof
of
the
above
claim
(∗
2
),
it
suffices
to
verify
that
Combinatorial
anabelian
topics
I
133
(†):
α
∈
Dehn(G
x
),
and,
moreover,
for
i
∈
{1,
2},
α
is
contained
in
the
kernel
of
the
natural
surjection
Dehn(G
x
)
Dehn(((G
x
)|
H
i
)
S
i
)
[cf.
Theorem
4.8,
(iii),
(iv)].
F
F
)
=
{e
F
Indeed,
since
[as
is
easily
verified]
Node(G
x
)
=
N
(v
new,x
1
,
e
2
},
it
follows
immediately
from
Theorem
4.8,
(iii),
(iv),
that
2
Ker
Dehn(G
x
)
Dehn(((G
x
)|
H
i
)
S
i
)
=
{1}
.
i=1
In
particular,
the
implication
(†)
⇒
(∗
2
)
holds.
The
remainder
of
the
proof
of
the
claim
(∗
2
)
is
devoted
to
verifying
the
above
(†).
∼
For
i
∈
{1,
2},
let
Π
(v
i
)
F
x
⊆
Π
G
x
←
Π
F
be
a
verticial
subgroup
of
∼
F
F
Π
G
x
←
Π
F
associated
to
the
vertex
(v
i
)
F
x
∈
V(e
i
)
\
{v
new,x
}.
Then
preserves
the
Π
F
-conjugacy
since
α
∈
Aut
|grph|
(G
x
),
it
follows
that
α
∼
class
of
Π
(v
i
)
F
x
⊆
Π
G
x
←
Π
F
.
Thus,
since
the
image
of
the
composite
Π
(v
i
)
F
x
→
Π
F
Π
F
/N
diag
is
commensurably
terminal
[cf.
Lemma
6.6,
(ii)],
it
follows
immediately
from
(a)
that
there
exists
an
N
diag
-conjugate
such
that
β
i
(Π
(v
i
)
F
x
)
=
Π
(v
i
)
F
x
.
β
i
[which
may
depend
on
i
∈
{1,
2}!]
of
α
Therefore,
since
the
composite
Π
(v
i
)
F
x
→
Π
F
Π
F
/N
diag
is
injective
[cf.
Lemma
6.6,
(iii)],
it
follows
from
(a)
that
β
i
induces
the
identity
automorphism
of
Π
(v
i
)
F
x
.
∼
Next,
let
Π
e
F
i
⊆
Π
(v
i
)
F
x
be
a
nodal
subgroup
of
Π
G
x
←
Π
F
associated
∼
F
;
Π
v
new,x
to
e
F
;i
⊆
Π
G
x
←
Π
F
i
∈
Node(G
x
)
that
is
contained
in
Π
(v
i
)
F
x
a
verticial
subgroup
[which
may
depend
on
i
∈
{1,
2}!]
associated
to
F
∈
Vert(G
x
)
which
contains
Π
e
F
i
:
v
new,x
F
⊆
Π
(v
i
)
F
x
Π
v
new,x
;i
⊇
Π
e
F
i
⊆
∼
Π
G
x
←
Π
F
.
Then
since
β
i
preserves
and
induces
the
identity
automorphism
on
Π
(v
i
)
F
x
,
it
follows
from
the
inclusion
Π
e
F
i
⊆
Π
(v
i
)
F
x
that
β
i
(Π
e
F
i
)
=
Π
e
F
i
.
More-
∼
F
over,
since
Π
v
new,x
;i
is
the
unique
verticial
subgroup
of
Π
G
x
←
Π
F
F
associated
to
v
new,x
which
contains
Π
e
F
i
[cf.
[CmbGC],
Proposition
1.5,
(i)],
it
follows
immediately
from
the
fact
that
α
∈
Aut
|grph|
(G
x
)
F
F
that
β
i
(Π
v
new,x
;i
)
=
Π
v
new,x
;i
.
Thus,
β
i
preserves
the
closed
subgroup
Π
F
i
⊆
Π
F
of
Π
F
obtained
by
forming
the
image
of
the
natural
homo-
morphism
F
→
Π
(v
i
)
F
x
−→
Π
F
lim
Π
v
new,x
;i
←
Π
e
F
i
−→
134
Yuichiro
Hoshi
and
Shinichi
Mochizuki
—
where
the
inductive
limit
is
taken
in
the
category
of
pro-Σ
F
groups.
Now
one
may
verify
easily
that
the
Π
F
-conjugacy
class
of
Π
F
i
⊆
Π
F
coincides
with
the
Π
F
-conjugacy
class
of
the
image
of
the
natural
outer
∼
injection
Π
((G
x
)|
H
i
)
Si
→
Π
G
x
←
Π
F
discussed
in
Proposition
2.11;
in
particular,
Π
F
i
is
commensurably
terminal
in
Π
F
[cf.
Proposition
2.11].
Moreover,
by
applying
a
similar
argument
to
the
argument
used
in
[CmbCsp],
Definition
2.1,
(iii),
(vi),
or
[NodNon],
Definition
5.1,
(ix),
(x)
[i.e.,
by
considering
the
portion
of
the
underlying
scheme
X
2
of
X
2
log
corresponding
to
the
underlying
scheme
(X
v
i
)
2
of
the
2-nd
log
configu-
ration
space
(X
v
i
)
log
of
the
stable
log
curve
X
v
log
determined
by
G|
v
i
],
2
i
∼
one
concludes
that
there
exists
a
verticial
subgroup
Π
v
i
⊆
Π
G
←
Π
B
associated
to
v
i
∈
Vert(G)
such
that
the
outer
representation
of
Π
v
i
on
ρ
2/1
Π
F
determined
by
the
composite
Π
v
i
→
Π
B
→
Out(Π
F
)
preserves
the
Π
F
-conjugacy
class
of
Π
F
i
⊆
Π
F
[so
we
obtain
a
natural
outer
repre-
sentation
Π
v
i
→
Out(Π
F
i
)
—
cf.
Lemma
2.12,
(iii)],
and,
moreover,
out
def
that
if
we
write
Π
T
i
=
Π
F
i
Π
v
i
(⊆
Π
T
)
[cf.
the
discussion
enti-
tled
“Topological
groups”
in
§0],
then
Π
T
i
is
naturally
isomorphic
to
the
“Π
T
”
obtained
by
taking
“G”
to
be
G|
v
i
.
Now
since
β
i
(Π
F
i
)
=
Π
F
i
,
and
α
∈
Z
Out
C
(Π
F
)
(ρ
2/1
(H)),
one
may
verify
easily
that
the
outomorphism
of
Π
F
i
determined
by
β
i
|
Π
F
i
[cf.
Lemma
2.12,
(iii)]
is
∈
Z
Out
C
(Π
F
i
)
(ρ
2/1
(H
∩
Π
v
i
))
—
where,
by
abuse
of
notation,
we
write
H
∩
Π
v
i
⊆
Π
B
for
the
intersection
of
H
with
the
image
of
Π
v
i
in
Π
B
.
Therefore,
since
the
quantity
“3g
−
3
+
r”
associated
to
G|
v
i
is
<
3g
−
3
+
r,
by
considering
a
similar
diagram
to
the
diagram
in
[CmbCsp],
Definition
2.1,
(vi),
or
[NodNon],
Definition
5.1,
(x),
and
applying
the
induction
hypothesis,
we
conclude
that
β
i
|
Π
F
i
is
a
Π
F
i
-inner
automorphism.
In
particular,
it
follows
immediately
[by
allowing
i
∈
{1,
2}
to
vary]
that
the
outomorphism
α
is
∈
Dehn(G
x
),
and,
moreover
—
by
considering
the
natural
identification
outer
isomorphism
∼
Π
F
i
→
Π
((G
x
)|
H
i
))
Si
—
that
α
is
contained
in
the
kernel
of
the
natural
surjection
Dehn(G
x
)
Dehn(((G
x
)|
H
i
))
S
i
),
as
desired.
This
completes
the
proof
of
(†),
hence
also
of
Lemma
6.8.
Q.E.D.
Definition
6.9.
In
the
notation
of
Definition
6.3:
(i)
Suppose
that
2g
−
2
+
r
>
1,
i.e.,
(g,
r)
∈
{(0,
3),
(1,
1)}.
Then
we
shall
write
def
A
g,r
=
{1}
⊆
Aut(Cusp
F
(G))
Combinatorial
anabelian
topics
I
135
[cf.
Definition
6.5,
(i)].
(ii)
Suppose
that
(g,
r)
=
(1,
1).
Then
we
shall
write
(Z/2Z
≃)
(iii)
def
A
g,r
=
Aut(Cusp
F
(G))
.
Suppose
that
(g,
r)
=
(0,
3).
Then
we
shall
write
(Z/2Z
×
Z/2Z
≃)
A
g,r
⊆
Aut(Cusp
F
(G))
for
the
subgroup
of
Aut(Cusp
F
(G))
obtained
as
the
image
of
the
subgroup
of
the
symmetric
group
on
4
letters
{id,
(1
2)(3
4),
(1
3)(2
4),
(1
4)(2
3)}
⊆
S
4
∼
via
the
isomorphism
S
4
→
Aut(Cusp
F
(G))
arising
from
a
bijec-
∼
tion
{1,
2,
3,
4}
→
Cusp
F
(G).
[Note
that
since
the
above
sub-
group
of
S
4
is
normal,
the
subgroup
A
g,r
⊆
Aut(Cusp
F
(G))
∼
does
not
depend
on
the
choice
of
the
bijection
{1,
2,
3,
4}
→
Cusp
F
(G).]
Lemma
6.10
(Permutations
of
cusps
arising
from
certain
C-admissible
outomorphisms).
In
the
notation
of
Definition
6.3,
let
H
⊆
Π
B
be
an
open
subgroup
of
Π
B
.
Then
the
following
hold:
(i)
The
composite
Z
Out
C
(Π
F
)
(ρ
2/1
(H))
→
Out
C
(Π
F
)
→
Aut(Cusp
F
(G))
[cf.
Definition
6.5,
(ii)]
factors
through
the
subgroup
A
g,r
⊆
Aut(Cusp
F
(G))
[cf.
Definition
6.9],
hence
determines
a
homo-
morphism
Z
Out
C
(Π
F
)
(Im(ρ
2/1
))
−→
A
g,r
.
(ii)
The
composite
Aut
X
log
(X
2
log
)
−→
Z
Out
C
(Π
F
)
(Im(ρ
2/1
))
−→
A
g,r
of
the
natural
homomorphism
Aut
X
log
(X
2
log
)
−→
Z
Out
C
(Π
F
)
(Im(ρ
2/1
))
136
Yuichiro
Hoshi
and
Shinichi
Mochizuki
with
the
homomorphism
of
(i)
is
an
isomorphism.
In
par-
ticular,
the
homomorphism
Z
Out
C
(Π
F
)
(Im(ρ
2/1
))
→
A
g,r
of
(i)
is
a
split
surjection
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Proof.
First,
we
verify
assertion
(i).
If
(g,
r)
=
(1,
1),
then
since
A
g,r
=
Aut(Cusp
F
(G)),
assertion
(i)
is
immediate.
On
the
other
hand,
if
r
=
0,
then
since
Cusp
F
(G)
=
1,
assertion
(i)
is
immediate.
Thus,
in
the
remainder
of
the
proof
of
assertion
(i),
we
suppose
that
(g,
r)
=
(1,
1),
r
≥
1.
Now
we
verify
assertion
(i)
in
the
case
where
r
=
1.
Let
us
observe
that
it
follows
immediately
from
Lemma
6.2,
(i),
that
by
replacing
X
log
by
a
suitable
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
,
we
may
as-
sume
without
loss
of
generality
[cf.
our
assumption
that
r
=
1,
which
im-
plies
that
(g,
r)
=
(0,
3)]
that
G
is
cyclically
primitive
[cf.
Definition
4.1].
Let
c
∈
Cusp(G)
be
the
unique
cusp
of
G,
e
∈
Node(G)
the
unique
node
of
G,
x
∈
X(k)
such
that
x
e,
and
α
∈
Z
Out
C
(Π
F
)
(ρ
2/1
(H)).
Then
let
us
observe
that
it
follows
immediately
from
our
assumption
that
G
is
cyclically
primitive
of
type
(g,
r)
=
(1,
1)
(respectively,
the
various
defi-
nitions
involved)
that
the
vertex
of
G
x
to
which
c
F
(respectively,
c
F
diag
)
abuts
is
not
of
type
(0,
3)
(respectively,
is
of
type
(0,
3)).
Moreover,
it
follows
immediately
from
Lemma
6.7,
(ii),
that
the
outomorphism
α
of
∼
Π
G
x
←
Π
F
is
∈
Aut(G
x
).
Thus,
we
conclude
that
the
automorphism
of
Cusp
F
(G)
induced
by
α
is
the
identity
automorphism.
This
completes
the
proof
of
assertion
(i)
in
the
case
where
r
=
1.
Next,
we
verify
assertion
(i)
in
the
case
where
r
>
1.
Let
us
observe
that
it
follows
immediately
from
Lemma
6.2,
(i),
that
by
replacing
X
log
by
a
suitable
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
,
we
may
assume
without
loss
of
generality
that
Node(G)
=
∅.
Let
v
∈
Vert(G)
be
the
unique
vertex
of
G
[cf.
our
assumption
that
Node(G)
=
∅]
and
α
∈
Z
Out
C
(Π
F
)
(ρ
2/1
(H)).
Now
let
us
observe
that
for
any
c
∈
Cusp(G),
x
∈
X(k)
such
that
x
c,
it
follows
immediately
from
the
various
def-
F
F
};
C(v
new,x
)
=
{c
F
,
c
F
initions
involved
that
Vert(G
x
)
=
{v
x
F
,
v
new,x
diag
};
F
F
F
F
F
C(v
x
)
=
Cusp(G
x
)\{c
,
c
diag
};
v
x
is
of
type
(g,
r);
v
new,x
is
of
type
(0,
3).
Moreover,
it
follows
immediately
from
Lemma
6.7,
(ii),
that
the
outo-
∼
morphism
α
of
Π
G
x
←
Π
F
is
∈
Aut(G
x
).
Thus,
if
(g,
r)
=
(0,
3),
then
F
F
is
of
type
(0,
3),
it
follows
imme-
since
v
x
is
of
type
(g,
r),
and
v
new,x
diately
that
α
induces
the
identity
automorphism
on
Vert(G
x
),
hence
F
that
α
preserves
the
subset
{c,
c
F
diag
}
⊆
Cusp
(G)
corresponding
to
F
C(v
new,x
)
=
{c
F
,
c
F
diag
}.
In
particular,
if
(g,
r)
=
(0,
3),
(respectively,
(g,
r)
=
(0,
3)),
then
—
by
allowing
“c”
to
vary
among
the
elements
of
Combinatorial
anabelian
topics
I
137
Cusp(G)
—
one
may
verify
easily
that
the
automorphism
of
Cusp
F
(G)
induced
by
α
is
the
identity
automorphism
(respectively,
satisfies
the
condition
that
for
any
subset
S
∈
Cusp
F
(G)
of
cardinality
2,
the
au-
tomorphism
of
Cusp
F
(G)
induced
by
α
determines
an
automorphism
of
the
set
{S,
Cusp
F
(G)
\
S}
⊆
F
2
Cusp
(G)
,
hence,
by
Lemma
6.11
below,
is
contained
in
A
g,r
⊆
Aut(Cusp
F
(G))).
This
completes
the
proof
of
assertion
(i)
in
the
case
where
r
>
1,
hence
also
of
assertion
(i).
Next,
we
verify
assertion
(ii).
One
verifies
easily
that
the
composite
of
natural
homomorphisms
∼
Aut
X
log
(X
2
log
)
→
Aut
Π
B
(Π
T
)/Inn(Π
F
)
→
Z
Out(Π
F
)
(Im(ρ
2/1
))
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
factors
through
Z
Out
C
(Π
F
)
(Im(ρ
2/1
))
⊆
Z
Out(Π
F
)
(Im(ρ
2/1
)).
In
particular,
we
obtain
a
natural
homomorphism
Aut
X
log
(X
2
log
)
→
Z
Out
C
(Π
F
)
(Im(ρ
2/1
)).
Now
the
fact
that
the
composite
Aut
X
log
(X
2
log
)
→
Z
Out
C
(Π
F
)
(Im(ρ
2/1
))
→
Out
C
(Π
F
)
→
Aut(Cusp
F
(G))
determines
a
surjection
Aut
X
log
(X
2
log
)
A
g,r
is
well-known
and
easily
verified.
To
verify
that
this
surjection
is
injective,
observe
that
an
el-
ement
of
the
kernel
of
this
surjection
determines
an
automorphism
of
the
trivial
family
X
log
×
(Spec
k)
log
X
log
→
X
log
over
X
log
that
preserves
the
image
of
the
diagonal.
On
the
other
hand,
since
the
relative
tangent
bundle
of
this
trivial
family
has
no
nonzero
global
sections,
one
con-
cludes
immediately
that
such
an
automorphism
is
constant,
i.e.,
arises
from
a
single
automorphism
of
the
fiber
X
log
over
(Spec
k)
log
that
is
compatible
with
the
diagonal,
hence
[as
is
easily
verified]
equal
to
the
identity
automorphism,
as
desired.
This
completes
the
proof
of
asser-
tion
(ii).
Q.E.D.
Lemma
6.11
(A
subgroup
of
the
symmetric
group
on
4
let-
ters).
Write
G
⊆
S
4
for
the
subgroup
of
the
symmetric
group
on
4
letters
S
4
consisting
of
g
∈
S
4
such
that
(∗):
for
any
subset
S
⊆
{1,
2,
3,
4}
of
cardinality
2,
the
automorphism
g
of
{1,
2,
3,
4}
determines
an
au-
tomorphism
of
the
set
{S,
{1,
2,
3,
4}
\
S}
⊆
2
{1,2,3,4}
.
138
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Then
G
=
{id,
(1
2)(3
4),
(1
3)(2
4),
(1
4)(2
3)}
.
Proof.
First,
let
us
observe
that
one
may
verify
easily
that
{id,
(1
2)(3
4),
(1
3)(2
4),
(1
4)(2
3)}
⊆
G
.
Thus,
to
verify
Lemma
6.11,
it
suffices
to
verify
that
G
=
4.
Next,
let
us
observe
that
it
follows
immediately
from
the
condition
(∗)
that
for
any
element
g
∈
G,
it
holds
that
g
4
=
id;
in
particular,
by
the
Sylow
Theorem,
together
with
the
fact
that
S
4
=
2
3
·
3,
we
conclude
that
G
is
a
2-group.
Thus,
to
verify
Lemma
6.11,
it
suffices
to
verify
that
G
=
8.
Next,
let
us
observe
that
it
follows
immediately
from
the
condition
(∗)
that
G
⊆
S
4
is
normal.
Thus,
if
G
=
8,
then
since
S
4
=
2
3
·
3,
and
(1
2)
∈
S
4
is
of
order
2,
again
by
the
Sylow
Theorem,
we
conclude
that
(1
2)
∈
G,
in
contradiction
to
the
fact
that
(1
2)
does
not
satisfy
the
condition
(∗).
This
completes
the
proof
of
Lemma
6.11.
Q.E.D.
Theorem
6.12
(Centralizers
of
geometric
monodromy
groups
arising
from
configuration
spaces).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
0
<
m
<
n
positive
integers;
Σ
F
⊆
Σ
B
nonempty
sets
of
prime
numbers;
k
an
algebraically
closed
field
of
characteristic
zero;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
given
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
Suppose
that
Σ
F
⊆
Σ
B
satisfy
one
of
the
following
two
conditions:
(1)
Σ
F
and
Σ
B
determine
PT-formations
[i.e.,
are
either
of
car-
dinality
one
or
equal
to
Primes
—
cf.
[MT],
Definition
1.1,
(ii)].
(2)
n
−
m
=
1
and
Σ
B
=
Primes.
Write
log
X
n
log
,
X
m
for
the
n-th,
m-th
log
configuration
spaces
of
the
stable
log
curve
X
log
def
[cf.
the
discussion
entitled
“Curves”
in
§0],
respectively;
Π
n
,
Π
B
=
Π
m
for
the
respective
maximal
pro-Σ
B
quotients
of
the
kernels
of
the
natu-
log
)
π
1
((Spec
k)
log
);
ral
surjections
π
1
(X
n
log
)
π
1
((Spec
k)
log
),
π
1
(X
m
Π
n/m
⊆
Π
n
for
the
kernel
of
the
surjection
Π
n
Π
B
=
Π
m
induced
log
obtained
by
forgetting
the
last
(n
−
m)
by
the
projection
X
n
log
→
X
m
Combinatorial
anabelian
topics
I
139
factors;
Π
F
for
the
maximal
pro-Σ
F
quotient
of
Π
n/m
;
Π
T
for
the
quo-
tient
of
Π
n
by
the
kernel
of
the
natural
surjection
Π
n/m
Π
F
.
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
F
−→
Π
T
−→
Π
B
−→
1
,
which
determines
an
outer
representation
ρ
n/m
:
Π
B
−→
Out(Π
F
)
.
Then
the
following
hold:
(i)
Let
H
⊆
Π
B
be
an
open
subgroup
of
Π
B
.
Recall
that
X
n
log
→
log
may
be
regarded
as
the
(n
−
m)-th
log
configuration
space
X
m
log
log
log
of
the
family
of
stable
log
curves
X
m+1
→
X
m
over
X
m
.
Then
the
composite
of
natural
homomorphisms
log
log
Aut
X
m
log
(X
log
(X
n
)
−→
Aut
Π
B
(Π
T
)/Inn(Π
F
)
m+1
)
−→
Aut
X
m
∼
−→
Z
Out(Π
F
)
(Im(ρ
n/m
))
⊆
Z
Out(Π
F
)
(ρ
n/m
(H))
—
where
the
first
arrow
is
the
homomorphism
arising
from
the
functoriality
of
the
construction
of
the
log
configuration
space;
the
third
arrow
is
the
isomorphism
appearing
in
the
discussion
entitled
“Topological
groups”
in
§0
—
determines
an
isomor-
phism
∼
log
Aut
X
m
log
(X
m+1
)
−→
Z
Out
FC
(Π
F
)
(ρ
n/m
(H))
—
where
we
write
Out
FC
(Π
F
)
for
the
group
of
FC-admissible
[cf.
Definition
6.1;
[CmbCsp],
Definition
1.1,
(ii)]
outomor-
phisms
of
Π
F
[cf.
Lemma
6.2,
(ii)].
Here,
we
recall
that
the
log
automorphism
group
Aut
X
m
log
(X
m+1
)
is
isomorphic
to
⎧
if
(g,
r,
m)
=
(0,
3,
1);
⎨
Z/2Z
×
Z/2Z
Z/2Z
if
(g,
r,
m)
=
(1,
1,
1);
⎩
{1}
if
(g,
r,
m)
∈
{(0,
3,
1),
(1,
1,
1)}.
(ii)
The
isomorphism
of
(i)
and
the
natural
inclusion
S
n−m
→
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
—
where
we
write
Out
PFC
(Π
F
)
for
the
group
of
PFC-admissible
[cf.
Definitions
1.4,
(iii);
6.1]
out-
omorphisms
of
Π
F
[cf.
Lemma
6.2,
(ii)]
—
determine
an
iso-
morphism
∼
log
Aut
X
m
log
(X
m+1
)
×
S
n−m
−→
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
.
140
(iii)
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Let
H
be
a
closed
subgroup
of
Out
PFC
(Π
F
)
that
contains
an
open
subgroup
of
Im(ρ
n/m
)
⊆
Out(Π
F
).
Then
H
is
almost
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
If,
moreover,
H
⊆
Out
FC
(Π
F
),
and
(g,
r,
m)
∈
{(0,
3,
1),
(1,
1,
1)},
then
H
is
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Proof.
First,
we
verify
assertion
(i).
We
begin
by
observing
that
log
the
description
of
the
automorphism
group
Aut
X
m
log
(X
m+1
)
given
in
the
statement
of
assertion
(i)
follows
immediately
from
Lemma
6.10,
(ii).
Next,
let
us
observe
that
(∗
1
):
to
verify
assertion
(i),
it
suffices
to
verify
asser-
tion
(i)
in
the
case
where
Σ
B
=
Primes.
Indeed,
this
follows
immediately
from
the
various
definitions
involved.
Thus,
in
the
remainder
of
the
proof
of
assertion
(i),
we
suppose
that
Σ
B
=
Primes.
Next,
we
claim
that
(∗
2
):
the
composite
homomorphism
of
assertion
(i)
determines
an
injection
log
Aut
X
m
log
(X
m+1
)
→
Z
Out
FC
(Π
F
)
(ρ
n/m
(H))
.
Indeed,
one
verifies
easily
that
the
composite
as
in
assertion
(i)
factors
through
Z
Out
FC
(Π
F
)
(ρ
n/m
(H)).
On
the
other
hand,
by
considering
the
log
action
of
Aut
X
m
log
(X
m+1
)
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
suitable
subquotients
[arising
from
fiber
subgroups]
of
Π
F
,
it
follows
immediately
that
the
composite
as
in
assertion
(i)
is
injective
[cf.
Lemma
6.10,
(ii)].
This
completes
the
proof
of
the
claim
(∗
2
).
Next,
we
claim
that
(∗
3
):
the
injection
of
(∗
2
)
is
an
isomorphism.
Indeed,
it
follows
immediately
from
the
various
definitions
involved
that
if
N
B
⊆
Π
B
is
a
fiber
subgroup
of
Π
B
of
length
1
[cf.
Lemma
6.2,
(ii);
[MT],
Definition
2.3,
(iii)],
then
the
natural
surjection
Π
T
×
Π
B
N
B
N
B
may
be
regarded
as
the
“Π
T
Π
B
”
obtained
by
taking
“(g,
r,
m,
n)”
to
be
(g,
r
+
m
−
1,
1,
n
−
m
+
1).
Thus,
by
applying
the
inclusion
Z
Out
FC
(Π
F
)
(ρ
n/m
(H))
⊆
Z
Out
FC
(Π
F
)
(ρ
n/m
(H
∩
N
B
))
and
replacing
Π
T
Π
B
by
Π
T
×
Π
B
N
B
N
B
,
we
may
assume
without
loss
of
generality
that
m
=
1.
On
the
other
hand,
it
follows
immediately
from
the
various
definitions
involved
that
if
N
F
⊆
Π
F
is
a
fiber
subgroup
of
Π
F
of
length
n
−
2,
then
the
natural
surjection
Π
T
/N
F
Π
B
may
Combinatorial
anabelian
topics
I
141
be
regarded
as
the
“Π
T
Π
B
”
obtained
by
taking
“(g,
r,
m,
n)”
to
be
(g,
r,
1,
2).
Thus,
since
the
natural
homomorphism
Out
FC
(Π
F
)
→
Out
FC
(Π
F
/N
F
)
is
injective
[cf.
[NodNon],
Theorem
B],
by
replacing
Π
T
Π
B
by
Π
T
/N
F
Π
B
,
we
may
assume
without
loss
of
generality
that
(m,
n)
=
(1,
2).
In
particular
—
in
light
of
our
assumption
that
Σ
B
=
Primes
[cf.
(∗
1
)]
—
we
are
in
the
situation
of
Definition
6.3.
Let
α
∈
Z
Out
FC
(Π
F
)
(ρ
n/m
(H)).
Then
it
follows
immediately
from
Lemma
6.10,
(ii),
that
there
exists
an
element
β
of
the
image
of
the
injection
of
(∗
2
)
such
that
α
◦
β
∈
Z
Out
FC
(Π
F
)
(ρ
n/m
(H))
induces
the
identity
automorphism
of
Cusp
F
(G)
[cf.
Definition
6.5,
(i),
(ii)].
In
par-
ticular,
α
◦
β
preserves
the
Π
F
-conjugacy
class
of
a
cuspidal
subgroup
F
Π
c
Fdiag
⊆
Π
F
of
Π
F
associated
to
c
F
diag
∈
Cusp
(G)
[cf.
Definition
6.5,
(iii)].
Thus,
it
follows
from
Lemma
6.8
that
α
◦
β
is
the
identity
outo-
morphism
of
Π
F
.
In
particular,
we
conclude
that
the
injection
of
(∗
2
)
is
surjective.
This
completes
the
proof
of
the
claim
(∗
3
),
hence
also
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
let
us
observe
that
by
consider-
ing
the
action
of
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
on
the
set
of
fiber
subgroups
of
Π
F
of
length
1,
we
obtain
an
exact
sequence
of
profinite
groups
1
−→
Z
Out
FC
(Π
F
)
(ρ
n/m
(H))
−→
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
−→
S
n−m
.
log
obtained
by
Now
by
considering
the
action
of
S
n−m
on
X
n
log
over
X
m
log
permuting
the
first
n
−
m
factors
of
X
n
,
we
obtain
a
section
S
n−m
→
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
of
the
third
arrow
in
the
above
exact
sequence;
in
particular,
the
third
arrow
is
surjective.
On
the
other
hand,
it
follows
from
[NodNon],
Theorem
B,
that
the
image
of
the
section
S
n−m
→
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
commutes
with
Z
Out
FC
(Π
F
)
(ρ
n/m
(H)).
Thus,
the
composite
of
natural
homomorphisms
∼
log
Aut
X
m
log
(X
m+1
)
→
Z
Out
FC
(Π
F
)
(ρ
n/m
(H))
→
Z
Out
FPC
(Π
F
)
(ρ
n/m
(H))
[cf.
assertion
(i)]
and
the
section
S
n−m
→
Z
Out
PFC
(Π
F
)
(ρ
n/m
(H))
de-
termine
an
isomorphism
as
in
the
statement
of
assertion
(ii).
This
com-
pletes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
assertions,
(i),
(ii).
This
completes
the
proof
of
Theorem
6.12.
Q.E.D.
Remark
6.12.1.
By
considering
a
suitable
specialization
isomor-
phism,
one
may
replace
the
expression
“k
an
algebraically
closed
field
of
characteristic
zero”
in
the
statement
of
Theorem
6.12
by
the
expression
“k
an
algebraically
closed
field
of
characteristic
∈
Σ
B
”.
142
Yuichiro
Hoshi
and
Shinichi
Mochizuki
Theorem
6.13
(Centralizers
of
geometric
monodromy
groups
arising
from
moduli
stacks
of
pointed
curves).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
Σ
a
nonempty
set
of
prime
numbers;
k
an
algebraically
closed
field
of
characteristic
def
zero.
Write
Π
M
g,r
=
π
1
((M
g,r
)
k
)
for
the
étale
fundamental
group
of
the
moduli
stack
(M
g,r
)
k
[cf.
the
discussion
entitled
“Curves”
in
§0];
Π
g,r
for
the
maximal
pro-Σ
quotient
of
the
kernel
N
g,r
of
the
natural
sur-
jection
π
1
((C
g,r
)
k
)
π
1
((M
g,r
)
k
)
=
Π
M
g,r
[cf.
the
discussion
entitled
“Curves”
in
§0];
Π
C
g,r
for
the
quotient
of
the
étale
fundamental
group
π
1
((C
g,r
)
k
)
of
(C
g,r
)
k
by
the
kernel
of
the
natural
surjection
N
g,r
Π
g,r
.
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
g,r
−→
Π
C
g,r
−→
Π
M
g,r
−→
1
,
which
determines
an
outer
representation
ρ
g,r
:
Π
M
g,r
−→
Out(Π
g,r
)
.
Then
the
following
hold:
(i)
The
profinite
group
Π
g,r
is
equipped
with
a
natural
structure
of
pro-Σ
surface
group
[cf.
[MT],
Definition
1.2].
(ii)
Let
H
⊆
Π
M
g,r
be
an
open
subgroup
of
Π
M
g,r
.
Suppose
that
2g
−
2
+
r
>
1,
i.e.,
(g,
r)
∈
{(0,
3),
(1,
1)}.
Then
the
composite
of
natural
homomorphisms
Aut
(M
g,r
)
k
((C
g,r
)
k
)
−→
Aut
Π
M
g,r
(Π
C
g,r
)/Inn(Π
g,r
)
∼
−→
Z
Out(Π
g,r
)
(Im(ρ
g,r
))
⊆
Z
Out(Π
g,r
)
(ρ
g,r
(H))
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
deter-
mines
an
isomorphism
∼
Aut
(M
g,r
)
k
((C
g,r
)
k
)
−→
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H))
[cf.
(i);
Definition
6.1].
Here,
we
recall
that
the
automorphism
group
Aut
(M
g,r
)
k
((C
g,r
)
k
)
is
isomorphic
to
⎧
if
(g,
r)
=
(0,
4);
⎨
Z/2Z
×
Z/2Z
Z/2Z
if
(g,
r)
∈
{(1,
2),
(2,
0)};
⎩
{1}
if
(g,
r)
∈
{(0,
4),
(1,
2),
(2,
0)}
.
Combinatorial
anabelian
topics
I
(iii)
143
Let
H
⊆
Out
C
(Π
g,r
)
be
a
closed
subgroup
of
Out
C
(Π
g,r
)
that
contains
an
open
subgroup
of
Im(ρ
g,r
)
⊆
Out(Π
g,r
).
Suppose
that
2g
−
2
+
r
>
1,
i.e.,
(g,
r)
∈
{(0,
3),
(1,
1)}.
Then
H
is
almost
slim
[cf.
the
discussion
entitled
“Topolog-
ical
groups”
in
§0].
If,
moreover,
2g
−
2
+
r
>
2,
i.e.,
(g,
r)
∈
{(0,
3),
(0,
4),
(1,
1),
(1,
2),
(2,
0)},
then
H
is
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Proof.
Assertion
(i)
follows
immediately
from
the
various
defini-
tions
involved.
Next,
we
verify
assertion
(ii).
First,
we
recall
that
the
description
of
the
automorphism
group
Aut
(M
g,r
)
k
((C
g,r
)
k
)
given
in
the
statement
of
assertion
(ii)
is
well-known
[cf.,
e.g.,
[CorHyp],
Theorem
B,
if
2g
−
2
+
r
>
2,
i.e.,
(g,
r)
∈
{(0,
4),
(1,
2),
(2,
0)}].
Next,
we
claim
that
(∗
1
):
the
composite
homomorphism
of
assertion
(ii)
determines
an
injection
Aut
(M
g,r
)
k
((C
g,r
)
k
)
→
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H))
.
Indeed,
one
verifies
easily
that
the
composite
as
in
assertion
(ii)
factors
through
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H)).
Thus,
the
claim
(∗
1
)
follows
immediately
from
the
well-known
fact
that
any
nontrivial
automorphism
of
a
hyper-
bolic
curve
over
an
algebraically
closed
field
of
characteristic
∈
Σ
induces
a
nontrivial
outomorphism
of
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
of
the
hyperbolic
curve
[cf.,
e.g.,
[LocAn],
the
proof
of
Theorem
14.1].
This
completes
the
proof
of
the
claim
(∗
1
).
Next,
we
claim
that
(∗
2
):
if
r
>
0,
then
the
injection
of
(∗
1
)
is
an
isomor-
phism.
Indeed,
write
N
⊆
Π
M
g,r
for
the
kernel
of
the
surjection
Π
M
g,r
π
1
((M
g,r−1
)
k
)
determined
by
the
(1-)morphism
(M
g,r
)
k
→
(M
g,r−1
)
k
obtained
by
forgetting
the
last
section.
Then
it
follows
immediately
from
the
various
definitions
involved
that
there
exists
a
commutative
diagram
of
profinite
groups
1
−−−−→
Π
g,r
−−−−→
E
−−−−→
N
−−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
1
−−−−→
Π
F
−−−−→
Π
T
−−−−→
Π
B
−−−−→
1
144
Yuichiro
Hoshi
and
Shinichi
Mochizuki
—
where
the
upper
sequence
is
the
exact
sequence
obtained
by
pulling
back
the
exact
sequence
1
→
Π
g,r
→
Π
C
g,r
→
Π
M
g,r
→
1
by
the
natural
inclusion
N
→
Π
M
g,r
;
the
lower
sequence
is
the
exact
sequence
“1
→
Π
F
→
Π
T
→
Π
B
→
1”
obtained
by
applying
the
procedure
given
in
the
statement
of
Theorem
6.12
in
the
case
where
(m,
n,
Σ
F
,
Σ
B
)
=
(1,
2,
Σ,
Primes)
to
a
stable
log
curve
of
type
(g,
r
−
1)
over
(Spec
k)
log
;
the
vertical
arrows
are
isomorphisms.
Thus,
it
follows
immediately
from
Theorem
6.12,
(i),
that
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H
∩
N
))
is
isomorphic
to
the
automorphism
group
Aut
X
log
(X
2
log
)
for
the
stable
log
curve
X
log
over
(Spec
k)
log
of
type
(g,
r
−
1).
In
particular,
by
the
claim
(∗
1
),
we
obtain
that
Aut
(M
g,r
)
k
((C
g,r
)
k
)
→
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H))
∼
⊆
Z
Out
C
(Π
g,r
)
(ρ
g,r
(H
∩
N
))
←
Aut
X
log
(X
2
log
)
.
Thus,
by
comparing
(Aut
(M
g,r
)
k
((C
g,r
)
k
))
with
Aut
X
log
(X
2
log
)
[cf.
The-
orem
6.12,
(i)],
we
conclude
that
the
injection
of
the
claim
(∗
1
)
is
an
isomorphism.
This
completes
the
proof
of
the
claim
(∗
2
).
Moreover,
it
follows
immediately
from
the
proof
of
the
claim
(∗
2
)
that
(∗
3
):
if
α
∈
Z
Out
C
(Π
0,4
)
(ρ
0,4
(H))
induces
the
identity
automorphism
on
the
set
of
conjugacy
classes
of
cus-
pidal
inertia
subgroups
of
Π
0,4
,
then
α
is
the
identity
outomorphism
of
Π
0,4
.
In
light
of
the
claim
(∗
2
),
in
the
remainder
of
the
proof
of
assertion
(ii),
we
assume
that
r
=
0,
hence
also
that
g
≥
2.
For
x
∈
(M
g,0
)
k
(k),
write
G
x
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
associated
to
the
log
log
log
def
geometric
fiber
of
(C
g,0
)
k
→
(M
g,0
)
k
over
x
log
=
x
×
(M
g,0
)
k
(M
g,0
)
k
;
thus,
we
have
a
natural
Im(ρ
g,0
)
(⊆
Out(Π
g,0
))-torsor
of
outer
isomor-
∼
∼
phisms
Π
g,0
→
Π
G
x
.
Let
us
fix
an
isomorphism
Π
g,0
→
Π
G
x
that
belongs
to
this
collection
of
isomorphisms.
Moreover,
for
x
∈
(M
g,0
)
k
(k),
we
shall
say
that
x
satisfies
the
condition
(†)
if
(†
1
)
Vert(G
x
)
=
{v
1
,
v
2
};
Node(G
x
)
=
{e
1
,
e
2
,
·
·
·
,
e
g+1
};
(†
2
)
N
(v
1
)
=
N
(v
2
)
=
Node(G
x
);
(†
3
)
v
1
and
v
2
are
of
type
(0,
g
+
1);
Combinatorial
anabelian
topics
I
145
we
shall
say
that
x
satisfies
the
condition
(‡)
if
(‡
1
)
Vert(G
x
)
=
{v
1
∗
,
v
2
∗
,
w
∗
};
Node(G
x
)
=
{e
∗
1
,
e
∗
2
,
·
·
·
,
e
∗
g+1
,
f
∗
};
(‡
2
)
N
(v
1
∗
)
=
{e
∗
1
,
e
∗
2
,
·
·
·
,
e
∗
g+1
};
N
(v
2
∗
)
=
{e
∗
1
,
e
∗
2
,
·
·
·
,
e
∗
g−1
,
f
∗
};
N
(w
∗
)
=
{e
∗
g
,
e
∗
g+1
,
f
∗
};
(‡
3
)
v
1
∗
is
of
type
(0,
g
+
1),
v
2
∗
is
of
type
(0,
g),
and
w
∗
is
of
type
(0,
3).
Let
us
observe
that
one
may
verify
easily
that
there
exists
a
k-valued
point
x
∈
(M
g,0
)
k
(k)
that
satisfies
(†);
if,
moreover,
g
>
2,
then
there
exists
a
k-valued
point
x
∈
(M
g,0
)
k
(k)
that
satisfies
(‡).
Let
x
∈
(M
g,0
)
k
(k)
be
a
k-valued
point.
Then
we
claim
that
(∗
4
):
if
x
satisfies
(†),
and,
relative
to
the
isomor-
∼
phism
Π
g,0
→
Π
G
x
fixed
above,
α
∈
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
determines
an
element
of
Aut
|grph|
(G
x
)
(⊆
Out(Π
G
x
)
∼
←
Out(Π
g,0
)),
then
for
any
e
∈
Node(G
x
),
the
image
α
e
of
α
via
the
natural
inclusion
Aut
|grph|
(G
x
)
→
Aut
|grph|
((G
x
)
{e}
)
[cf.
Proposition
2.9,
(ii)]
satisfies
α
e
∈
Dehn((G
x
)
{e}
)
.
Indeed,
let
e
∈
Node(G
x
)
and
y
∈
(M
g,0
)
k
(k)
a
k-valued
point
such
that
G
y
corresponds
to
(G
x
)
{e}
[cf.
the
special
fibers
of
the
stable
log
curves
over
“S
log
”
that
appear
in
Proposition
5.6,
(iii)].
Write
v
∈
Vert(G
y
)
for
the
unique
vertex
of
G
y
.
[Note
that
it
follows
from
the
definition
of
the
condition
(†)
that
Vert(G
y
)
=
1.]
Then
it
follows
immediately
from
the
general
theory
of
stable
log
curves
that
there
exist
a
“clutching
(1-)morphism”
corresponding
to
the
operation
of
resolving
the
nodes
of
G
y
[i.e.,
obtained
by
forming
appropriate
composites
of
the
clutching
morphisms
discussed
in
[Knud],
Definition
3.6]
(M
0,2g
)
k
−→
(M
g,0
)
k
and
a
k-valued
point
y
∈
(M
0,2g
)
k
(k)
such
that
the
image
of
y
via
the
above
clutching
morphism
coincides
with
y,
and,
moreover,
G
y
is
natu-
rally
isomorphic
to
(G
y
)|
v
.
Write
(M
log
0,2g
)
k
for
the
log
stack
obtained
by
equipping
(M
0,2g
)
k
with
the
log
structure
induced
by
the
log
structure
log
of
(M
g,0
)
k
via
the
above
clutching
morphism.
Then
one
verifies
easily
146
Yuichiro
Hoshi
and
Shinichi
Mochizuki
that
the
composite
∼
log
def
ρ
g,0
Π
M
0,2g
=
π
1
((M
log
0,2g
)
k
)
−→
π
1
((M
g,0
)
k
)
←−
Π
M
g,0
−→
Out(Π
g,0
)
—
where
the
first
arrow
is
the
outer
homomorphism
induced
by
the
above
clutching
morphism,
and
the
second
arrow
is
the
outer
isomor-
phism
obtained
by
applying
the
“log
purity
theorem”
to
the
natural
log
(1-)morphism
(M
g,0
)
k
→
(M
g,0
)
k
[cf.
[ExtFam],
Theorem
B]
—
fac-
∼
tors
through
Aut
|grph|
(G
y
)
⊆
Out(Π
G
y
)
←
Out(Π
g,0
).
Moreover,
the
resulting
homomorphism
Π
M
0,2g
→
Aut
|grph|
(G
y
)
fits
into
a
commuta-
tive
diagram
of
profinite
groups
Π
M
0,2g
⏐
⏐
−−−−→
Π
M
0,2g
⏐
⏐
ρ
Vert
G
y
Aut
|grph|
(G
y
)
−−−−→
Glu(G
y
)
=
Aut
|grph|
((G
y
)|
v
)
[cf.
Definition
4.9;
Proposition
4.10,
(ii)]
—
where
the
upper
hori-
zontal
arrow
is
the
outer
homomorphism
induced
by
the
(1-)morphism
(M
log
0,2g
)
k
→
(M
0,2g
)
k
obtained
by
forgetting
the
log
structure.
More-
over,
one
verifies
easily
that
there
exists
a
natural
outer
isomorphism
∼
Π
(G
y
)|
v
→
Π
0,2g
such
that
the
homomorphism
Π
M
0,2g
→
Out(Π
0,2g
)
ob-
tained
by
conjugating
the
outer
action
implicit
in
the
right-hand
vertical
arrow
of
the
above
diagram
Π
M
0,2g
→
Aut
|grph|
((G
y
)|
v
)
⊆
Out(Π
(G
y
)|
v
)
∼
by
the
outer
isomorphism
Π
(G
y
)|
v
→
Π
0,2g
coincides
with
ρ
0,2g
.
Thus,
by
considering
the
image
in
Π
M
0,2g
of
the
inverse
image
of
H
⊆
Π
M
g,0
in
Π
M
0,2g
[cf.
the
diagrams
of
the
above
displays],
it
follows
immedi-
ately
from
the
claims
(∗
2
)
[in
the
case
where
“(g,
r)”=
(0,
2g)]
and
(∗
3
)
[in
the
case
where
g
=
2],
together
with
the
various
definitions
involved,
that
if
α
∈
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
determines
an
element
of
Aut
|grph|
(G
x
)
∼
(⊆
Out(Π
G
x
)
←
Out(Π
g,0
)),
then
the
image
of
α
via
Aut
|grph|
(G
x
)
→
ρ
Vert
G
y
∼
Aut
|grph|
((G
x
)
{e}
)
→
Aut
|grph|
(G
y
)
Glu(G
y
)
=
Aut
|grph|
((G
y
)|
v
)
[cf.
Proposition
2.9,
(ii)]
is
trivial.
In
particular,
it
follows
from
Propo-
sition
4.10,
(ii),
that
the
image
α
e
of
α
via
Aut
|grph|
(G
x
)
→
Aut
|grph|
((G
x
)
{e}
)
satisfies
α
e
∈
Dehn((G
x
)
{e}
).
This
completes
the
proof
of
the
claim
(∗
4
).
Next,
we
claim
that
Combinatorial
anabelian
topics
I
147
(∗
5
):
if
x
satisfies
(†),
and
α
∈
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
determines
an
element
of
Aut
|grph|
(G
x
)
(⊆
Out(Π
G
x
)
∼
←
Out(Π
g,0
)),
then
α
is
the
identity
outomorphism
of
Π
g,0
.
Indeed,
it
follows
from
the
claim
(∗
4
)
that
Im
Dehn((G
x
)
{e}
)
→
Dehn(G
x
)
α
∈
e∈Node(G
x
)
[cf.
Theorem
4.8,
(ii)].
On
the
other
hand,
it
follows
immediately
from
Theorem
4.8,
(ii),
(iv),
that
the
right-hand
intersection
is
=
{1}.
This
completes
the
proof
of
the
claim
(∗
5
).
Next,
we
claim
that
(∗
6
):
we
have
∼
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
Aut
|Node(G
x
)|
(G
x
)
(⊆
Out(Π
G
x
)
←
Out(Π
g,0
))
;
if,
moreover,
x
satisfies
(‡),
then
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
Aut
|grph|
(G
x
)
.
Indeed,
it
follows
immediately
from
Proposition
5.6,
(ii),
together
with
log
the
definition
of
x
log
=
x
×
(M
g,0
)
k
(M
g,0
)
k
,
that
the
composite
log
∼
ρ
g,0
π
1
(x
log
)
−→
π
1
((M
g,0
)
k
)
←−
Π
M
g,0
−→
Out(Π
g,0
)
—
where
the
second
arrow
is
the
outer
isomorphism
obtained
by
apply-
ing
the
“log
purity
theorem”
to
the
natural
(1-)morphism
(M
g,0
)
k
→
log
(M
g,0
)
k
[cf.
[ExtFam],
Theorem
B]
—
determines
a
surjection
π
1
(x
log
)
∼
Dehn(G
x
)
(⊆
Out(Π
G
x
)
←
Out(Π
g,0
))
[i.e.,
which
induces
an
iso-
morphism
between
the
respective
maximal
pro-Σ
quotients].
Thus,
it
follows
immediately
from
the
various
definitions
involved
that
there
ex-
ists
an
open
subgroup
M
⊆
Dehn(G
x
)
such
that
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
∼
Z
Out
C
(Π
G
x
)
(M
)
relative
to
the
identification
Out
C
(Π
g,0
)
→
Out
C
(Π
G
x
)
∼
arising
from
our
choice
of
an
isomorphism
Π
g,0
→
Π
G
x
.
Therefore,
the
inclusion
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
Aut
|Node(G
x
)|
(G
x
)
follows
immedi-
ately
from
Theorem
5.14,
(ii).
This
completes
the
proof
of
the
inclusion
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
Aut
|Node(G
x
)|
(G
x
).
On
the
other
hand,
if,
more-
over,
x
satisfies
(‡),
then
it
follows
immediately
from
the
definition
of
the
condition
(‡)
that
Aut
|grph|
(G
x
)
=
Aut
|Node(G
x
)|
(G
x
).
In
particular,
148
Yuichiro
Hoshi
and
Shinichi
Mochizuki
we
obtain
that
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
Aut
|grph|
(G
x
).
This
completes
the
proof
of
the
claim
(∗
6
).
Next,
we
claim
that
(∗
7
):
if
x
satisfies
(†),
then
for
any
α
∈
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H)),
there
exists
an
element
β
of
the
image
of
the
injection
of
(∗
1
)
such
that
the
outo-
∼
morphism
α
◦
β
of
Π
g,0
→
Π
G
x
is
∈
Aut
|grph|
(G
x
)
∼
(⊆
Out(Π
G
x
)
←
Out(Π
g,0
)).
Indeed,
suppose
that
g
>
2.
Then
by
the
definitions
of
(†),
(‡),
one
may
verify
easily
that
there
exist
y
∈
(M
g,0
)
k
(k)
and
f
∈
Node(G
y
)
such
that
y
satisfies
(‡),
and,
moreover,
G
x
corresponds
to
(G
y
)
{f
}
[cf.
Proposition
5.6,
(iv)].
Thus,
it
follows
immediately
from
the
claim
(∗
6
)
that
Z
Out
C
(Π
g,0
)
(ρ
g,0
(H))
⊆
Aut
|grph|
(G
y
)
→
Aut
|grph|
(G
x
)
[cf.
Proposi-
tion
2.9,
(ii)],
i.e.,
so
we
may
take
β
to
be
the
identity
outomorphism.
This
completes
the
proof
of
the
claim
(∗
7
)
in
the
case
where
g
>
2.
Next,
suppose
that
g
=
2.
Write
G
x
for
the
underlying
semi-graph
of
G
x
and
Aut
|Node|
(G
x
)
for
the
group
of
automorphisms
of
G
x
which
induce
the
identity
automorphism
of
the
set
of
nodes
of
G
x
.
Then
one
may
verify
easily
from
the
explicit
structure
of
G
x
[cf.
the
definition
of
the
condition
(†)]
that
Aut
|Node|
(G
x
)
is
isomorphic
to
Z/2Z.
Thus,
since
the
automorphism
group
Aut
(M
2,0
)
k
((C
2,0
)
k
)
is
isomorphic
to
Z/2Z,
it
follows
immediately
from
the
claim
(∗
6
),
together
with
the
various
defi-
nitions
involved,
that
—
to
complete
the
proof
of
the
claim
(∗
7
)
in
the
case
where
g
=
2
—
it
suffices
to
verify
that
the
composite
of
natural
homomorphisms
Aut
(M
2,0
)
k
((C
2,0
)
k
)
−→
Aut(G
x
)
−→
Aut(G
x
)
factors
through
Aut
|Node|
(G
x
)
⊆
Aut(G
x
)
and
is
injective.
Now
the
fact
that
the
composite
in
question
factors
through
Aut
|Node|
(G
x
)
⊆
Aut(G
x
)
follows
immediately
from
the
claim
(∗
6
),
applied
to
elements
of
the
image
of
the
injection
of
(∗
1
).
On
the
other
hand,
the
injectivity
of
the
composite
in
question
follows
immediately
from
the
injectivity
of
the
natural
homomorphism
Aut
(M
2,0
)
k
((C
2,0
)
k
)
→
Aut(G
x
)
[cf.
the
proof
of
the
claim
(∗
1
)]
and
the
claim
(∗
5
).
This
completes
the
proof
of
the
claim
(∗
7
)
in
the
case
where
g
=
2,
hence
also
—
in
light
of
the
above
proof
of
the
claim
(∗
7
)
in
the
case
where
g
>
2
—
of
the
claim
(∗
7
).
Thus,
the
surjectivity
of
the
injection
of
(∗
1
)
follows
immediately
from
the
claims
(∗
5
)
and
(∗
7
).
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
assertion
(ii).
This
completes
the
proof
of
Theorem
6.13.
Q.E.D.
Combinatorial
anabelian
topics
I
149
Remark
6.13.1.
In
the
notation
of
Theorem
6.13,
since
Π
M
0,3
=
{1},
it
is
immediate
that
a
similar
result
to
the
results
stated
in
Theo-
rem
6.13,
(ii),
(iii),
does
not
hold
in
the
case
where
(g,
r)
=
(0,
3).
On
the
other
hand,
it
is
not
clear
to
the
authors
at
the
time
of
writing
whether
or
not
a
similar
result
to
the
results
stated
in
Theorem
6.13,
(ii),
(iii),
holds
in
the
case
where
(g,
r)
=
(1,
1).
Nevertheless,
we
are
able
to
obtain
a
conditional
result
concerning
the
centralizer
of
the
geometric
monodromy
group
in
the
case
where
(g,
r)
=
(1,
1)
[cf.
Theorem
6.14,
(iii),
(iv)
below].
Theorem
6.14
(Centralizers
of
geometric
monodromy
groups
arising
from
moduli
stacks
of
punctured
semi-ellptic
±
)
k
for
the
stack-
curves).
In
the
notation
of
Theorem
6.13,
write
(C
1,1
theoretic
quotient
of
(C
1,1
)
k
by
the
natural
action
of
Aut
(M
1,1
)
k
((C
1,1
)
k
)
over
the
moduli
stack
(M
1,1
)
k
;
Π
±
1,1
for
the
maximal
pro-Σ
quotient
±
±
of
the
kernel
N
1,1
=
Ker(π
1
((C
1,1
)
k
)
π
1
((M
1,1
)
k
)
=
Π
M
1,1
)
of
the
±
natural
surjection
π
1
((C
1,1
)
k
)
π
1
((M
1,1
)
k
)
=
Π
M
1,1
;
Π
C
±
for
the
def
1,1
±
±
)
k
)
of
the
stack
(C
1,1
)
k
quotient
of
the
étale
fundamental
group
π
1
((C
1,1
±
±
by
the
kernel
of
the
natural
surjection
N
1,1
Π
1,1
.
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
±
1,1
−→
Π
C
±
−→
Π
M
1,1
−→
1
,
1,1
which
determines
an
outer
representation
±
ρ
±
1,1
:
Π
M
1,1
−→
Out(Π
1,1
)
.
±
Write
Out
C
(Π
±
1,1
)
for
the
group
of
outomorphisms
of
Π
1,1
which
induce
±
bijections
on
the
set
of
cuspidal
inertia
subgroups
of
Π
1,1
.
Suppose
that
2
∈
Σ.
Then
the
following
hold:
(i)
The
profinite
group
Π
±
1,1
is
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
(ii)
Let
H
⊆
Π
M
1,1
be
an
open
subgroup
of
Π
M
1,1
.
Then
the
com-
posite
of
natural
homomorphisms
±
)
k
)
−→
Aut
Π
M
1,1
(Π
C
±
)/Inn(Π
±
Aut
(M
1,1
)
k
((C
1,1
1,1
)
1,1
∼
±
−→
Z
Out(Π
±
)
(Im(ρ
±
1,1
))
⊆
Z
Out(Π
±
)
(ρ
1,1
(H))
1,1
1,1
150
Yuichiro
Hoshi
and
Shinichi
Mochizuki
[cf.
(i);
the
discussion
entitled
“Topological
groups”
in
§0]
de-
termines
an
isomorphism
∼
±
)
k
)
−→
Z
Out
C
(Π
±
)
(ρ
±
Aut
(M
1,1
)
k
((C
1,1
1,1
(H))
.
1,1
±
Here,
we
recall
that
Aut
(M
1,1
)
k
((C
1,1
)
k
)
=
{1}.
(iii)
Let
H
⊆
Π
M
1,1
be
an
open
subgroup
of
Π
M
1,1
.
Then
the
com-
posite
of
natural
homomorphisms
Aut
(M
1,1
)
k
((C
1,1
)
k
)
−→
Aut
Π
M
1,1
(Π
C
1,1
)/Inn(Π
1,1
)
∼
−→
Z
Out(Π
1,1
)
(Im(ρ
1,1
))
⊆
Z
Out(Π
1,1
)
(ρ
1,1
(H))
[cf.
Theorem
6.13,
(i);
the
discussion
entitled
“Topological
groups”
in
§0]
determines
an
injection
Aut
(M
1,1
)
k
((C
1,1
)
k
)
→
Z
Out
C
(Π
1,1
)
(ρ
1,1
(H))
.
Moreover,
the
image
of
this
injection
is
centrally
terminal
in
Z
Out
C
(Π
1,1
)
(ρ
1,1
(H))
[cf.
the
discussion
entitled
“Topologi-
cal
groups”
in
§0].
Here,
we
recall
that
Aut
(M
1,1
)
k
((C
1,1
)
k
)
≃
Z/2Z.
(iv)
The
composite
of
natural
homomorphisms
Aut
(M
1,1
)
k
((C
1,1
)
k
)
−→
Aut
Π
M
1,1
(Π
C
1,1
)/Inn(Π
1,1
)
∼
−→
Z
Out(Π
1,1
)
(Im(ρ
1,1
))
[cf.
Theorem
6.13,
(i);
the
discussion
entitled
“Topological
groups”
in
§0]
determines
an
isomorphism
∼
Aut
(M
1,1
)
k
((C
1,1
)
k
)
−→
Z
Out
C
(Π
1,1
)
(Im(ρ
1,1
))
.
Proof.
Assertion
(i)
follows
immediately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
[MT],
Proposition
1.4.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
let
us
recall
that
the
description
±
)
k
)
given
in
the
statement
of
the
automorphism
group
Aut
(M
1,1
)
k
((C
1,1
of
assertion
(ii)
is
well-known
and
easily
verified.
Write
E
→
(M
1,1
)
k
for
the
family
of
elliptic
curves
determined
by
the
family
of
hyperbolic
curves
(C
1,1
)
k
→
(M
1,1
)
k
of
type
(1,
1);
U
→
(C
1,1
)
k
for
the
restriction
of
the
finite
étale
covering
E
→
E
over
(M
1,1
)
k
given
by
multiplica-
tion
by
2
to
(C
1,1
)
k
⊆
E.
Then
one
verifies
easily
that
the
action
of
Combinatorial
anabelian
topics
I
151
Aut
(M
1,1
)
k
((C
1,1
)
k
)
on
(C
1,1
)
k
lifts
naturally
to
an
action
[i.e.,
given
by
“multiplication
by
±1”]
on
U
over
(M
1,1
)
k
.
Write
P
for
the
stack-
theoretic
quotient
of
U
by
the
action
of
Aut
(M
1,1
)
k
((C
1,1
)
k
)
on
U;
Π
P/M
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
surjection
π
1
(P)
π
1
((M
1,1
)
k
);
ρ
P/M
:
Π
M
1,1
−→
Out(Π
P/M
)
for
the
natural
pro-Σ
outer
representation
arising
from
the
family
of
hyperbolic
curves
P
→
(M
1,1
)
k
.
Thus,
since
2
∈
Σ,
one
verifies
easily
that
Π
P/M
may
be
regarded
as
a
normal
open
subgroup
of
Π
±
1,1
.
Now
let
us
observe
that
one
verifies
easily
that
(∗
1
):
P
→
(M
1,1
)
k
is
a
family
of
hyperbolic
curves
of
type
(0,
4).
If,
moreover,
we
denote
by
T
→
(M
1,1
)
k
the
connected
finite
étale
covering
that
trivializes
the
finite
étale
covering
determined
by
the
four
cusps
of
P
→
(M
1,1
)
k
,
then
the
classifying
(1-)morphism
T
→
(M
0,4
)
k
of
P
×
(M
1,1
)
k
T
→
T
[which
is
well-
defined
up
to
the
natural
action
of
S
4
on
(M
0,4
)
k
]
is
dominant.
Now
we
claim
that
(∗
2
):
every
element
of
Out
C
(Π
±
1,1
)
preserves
the
nor-
mal
open
subgroup
Π
P/M
⊆
Π
±
1,1
.
Indeed,
let
us
observe
that
one
verifies
easily
that
the
natural
surjections
±
±
Π
±
1,1
Π
1,1
/Π
1,1
,
Π
1,1
/Π
P/M
determine
an
isomorphism
∼
±
ab
⊗
Z
Σ
Z/2Z
−→
(Π
±
(Π
±
1,1
)
1,1
/Π
1,1
)
×
(Π
1,1
/Π
P/M
)
.
Moreover,
it
follows
immediately
from
the
various
definitions
involved
ab
⊗
Z
Σ
Z/2Z
on
the
set
of
conjugacy
that
the
natural
action
of
(Π
±
1,1
)
classes
of
cuspidal
inertia
subgroups
of
the
kernel
of
the
natural
surjec-
±
ab
⊗
Z
Σ
Z/2Z
[which
is
equipped
with
a
natural
struc-
tion
Π
±
1,1
(Π
1,1
)
ab
ture
of
pro-Σ
surface
group
of
type
(1,
4)]
factors
through
(Π
±
⊗
Z
Σ
1,1
)
∼
pr
2
±
±
Z/2Z
→
(Π
±
1,1
/Π
1,1
)
×
(Π
1,1
/Π
P/M
)
(Π
1,1
/Π
P/M
),
and
that
the
re-
sulting
action
of
(Π
±
1,1
/Π
P/M
)
is
faithful.
Thus,
we
conclude
that
every
element
of
Out
C
(Π
±
1,1
)
preserves
the
normal
open
subgroup
Π
P/M
⊆
±
Π
1,1
.
This
completes
the
proof
of
the
claim
(∗
2
).
To
verify
assertion
(ii),
take
an
element
α
±
∈
Z
Out
C
(Π
±
)
(ρ
±
1,1
(H)).
1,1
±
Then
it
follows
from
the
claim
(∗
2
)
that
α
naturally
determines
an
ele-
C
ment
α
P
∈
Aut(Π
P/M
)/Inn(Π
±
1,1
).
Let
us
fix
a
lifting
β
∈
Out
(Π
P/M
)
152
Yuichiro
Hoshi
and
Shinichi
Mochizuki
of
α
P
.
Next,
let
us
observe
that
since
Π
±
1,1
/Π
P/M
is
finite,
to
ver-
ify
assertion
(ii),
by
replacing
H
by
an
open
subgroup
of
Π
M
1,1
con-
tained
in
H,
we
may
assume
without
loss
of
generality
that
β
com-
mutes
with
ρ
P/M
(H)
⊆
Out(Π
P/M
),
i.e.,
β
∈
Z
Out
C
(Π
P/M
)
(ρ
P/M
(H)).
Then
it
follows
immediately
from
Theorem
6.13,
(ii),
in
the
case
where
(g,
r)
=
(0,
4),
together
with
(∗
1
),
that
β
is
contained
in
the
image
of
the
natural
injection
Π
±
1,1
/Π
P/M
→
Out(Π
P/M
)
obtained
by
conjugation.
Thus,
α
P
,
hence
also
—
by
the
manifest
injectivity
[cf.
assertion
(i)]
±
of
the
homomorphism
Out
C
(Π
±
1,1
)
→
Aut(Π
P/M
)/Inn(Π
1,1
)
implicit
in
the
content
of
the
claim
(∗
2
)
—
α
±
,
is
trivial.
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
First,
recall
that
the
description
of
Aut
(M
1,1
)
k
((C
1,1
)
k
)
given
in
the
statement
of
assertion
(iii)
is
well-known
and
easily
verified.
Next,
let
us
observe
that
the
fact
that
the
composite
in
the
statement
of
assertion
(iii)
determines
an
injection
Aut
(M
1,1
)
k
((C
1,1
)
k
)
→
Z
Out
C
(Π
1,1
)
(ρ
1,1
(H))
follows
immediately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
the
claim
(∗
1
)
in
the
proof
of
Theorem
6.13,
(ii),
together
with
the
various
definitions
involved.
Next,
let
us
observe
that
by
applying
∼
out
the
natural
outer
isomorphism
Π
±
1,1
→
Π
1,1
Aut
(M
1,1
)
k
((C
1,1
)
k
),
we
obtain
an
exact
sequence
of
profinite
groups
1
−→
Aut
(M
1,1
)
k
((C
1,1
)
k
)
−→
Z
Out(Π
1,1
)
(Aut
(M
1,1
)
k
((C
1,1
)
k
))
−→
Out(Π
±
1,1
)
—
where
we
regard
Aut
(M
1,1
)
k
((C
1,1
)
k
)
as
a
closed
subgroup
of
Out(Π
1,1
)
by
means
of
the
injection
“
→”
of
the
above
display.
Thus,
the
central
terminality
asserted
in
the
statement
of
assertion
(iii)
follows
immedi-
ately,
in
light
of
the
above
exact
sequence,
from
assertion
(ii).
This
completes
the
proof
of
assertion
(iii).
Finally,
we
verify
assertion
(iv).
It
follows
immediately
from
asser-
tion
(iii)
that
the
image
of
the
homomorphism
Aut
(M
1,1
)
k
((C
1,1
)
k
)
→
Z
Out
C
(Π
1,1
)
(Im(ρ
1,1
))
determined
by
the
composite
in
the
statement
of
assertion
(iv)
is
centrally
terminal.
On
the
other
hand,
as
is
well-
known,
this
image
of
Aut
(M
1,1
)
k
((C
1,1
)
k
)
in
Out(Π
1,1
)
is
contained
in
Im(ρ
1,1
)
⊆
Out(Π
1,1
).
[Indeed,
recall
that
there
exists
a
natural
outer
∼
isomorphism
SL
2
(Z)
∧
→
Π
M
1,1
,
where
we
write
SL
2
(Z)
∧
for
the
profi-
−1
0
nite
completion
of
SL
2
(Z),
such
that
the
image
of
∈
SL
2
(Z)
∧
0
−1
Combinatorial
anabelian
topics
I
153
in
Out(Π
1,1
)
coincides
with
the
image
of
the
unique
nontrivial
element
of
Aut
(M
1,1
)
k
((C
1,1
)
k
)
≃
Z/2Z
in
Out(Π
1,1
).]
Now
assertion
(iv)
follows
immediately.
This
completes
the
proof
of
assertion
(iv).
Q.E.D.
Remark
6.14.1.
The
authors
hope
to
be
able
to
address
the
issue
of
whether
or
not
a
similar
result
to
the
results
stated
in
Theorem
6.13,
(ii),
(iii),
holds
for
other
families
of
pointed
curves
[e.g.,
the
universal
curves
over
moduli
stacks
of
hyperelliptic
curves
or
more
general
Hurwitz
stacks]
in
a
sequel
to
the
present
paper.
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(Yuichiro
Hoshi)
Research
Institute
for
Mathematical
Sciences,
Kyoto
University,
Kyoto
606-8502,
JAPAN
E-mail
address:
yuichiro@kurims.kyoto-u.ac.jp
(Shinichi
Mochizuki)
Research
Institute
for
Mathematical
Sciences,
Kyoto
Univer-
sity,
Kyoto
606-8502,
JAPAN
E-mail
address:
motizuki@kurims.kyoto-u.ac.jp